Method and apparatus for optimizing spectacle lenses, in particular for wearers of implanted intraocular lenses

ABSTRACT

A computer-implemented method for identifying relevant individual parameters of at least one eye of a spectacle wearer for the calculation or optimization of an ophthalmic lens for the at least one eye of the spectacle wearer, including: providing individual data on properties of the at least one eye of the spectacle wearer; constructing an individual eye model by defining a set of parameters of the individual eye model; and determining a probability distribution of values of the parameters of the individual eye model on the basis of the individual data.

TECHNICAL FIELD

The present invention relates to a method, a device, and a correspondingcomputer program product for determining relevant individual parametersof at least one eye of a spectacle wearer for the calculation oroptimization of a spectacle lens for the at least one eye of thespectacle wearer, and to a corresponding method, a device, and acomputer program product for the calculation (optimization) andmanufacture of a spectacle lens with the help of a partially individualeye model. In particular, the at least one eye of the spectacle wearerhas an implanted intraocular lens (IOL). For example, instead of or inaddition to the natural eye lens, an intraocular lens can have beenimplanted in the at least one eye during surgery. In other words, thespectacle wearer can in particular be a wearer of an implantedintraocular lens. Moreover, the present invention relates to a method, adevice and a corresponding computer program product for the calculation(optimization) and manufacture of a spectacle lens, in particular for awearer of an implanted intraocular lens, with the help of a partiallyindividual eye model.

BACKGROUND

For the manufacture or optimization of spectacle lenses, in particularindividual spectacle lenses, each spectacle lens is manufactured suchthat for each desired viewing direction or each desired object point thebest possible correction of a refractive error of the respective eye ofthe spectacle wearer is achieved. In general, a spectacle lens isconsidered to be fully correcting for a given viewing direction if thevalues for sphere, cylinder and axis of the wavefront upon passing thevertex sphere conform to the values for sphere, cylinder and axis of theprescription for the eye having the vision disorder. In the refractiondetermination for an eye of a spectacle wearer, dioptric values (inparticular sphere, cylinder, cylinder axis—i.e. in particularsphero-cylindrical deviations) for a far (usually infinite) distance andoptionally (for multifocal lenses or progressive lenses) an addition ora complete near refraction for a near distance (e.g. according to DIN58208) are determined. In the case of modern spectacle lenses, objectdistances deviating from the norm, which are used in the refractiondetermination, can also be specified. In this way, the prescription (inparticular sphere, cylinder, cylinder axis, and optionally addition ornear refraction) to be communicated to a lens manufacturer isstipulated. Knowledge of a special or individual anatomy of therespective eye or the refractive values of the eye having the visiondisorder, which are actually present in the individual case, is notrequired here.

However, a full correction for all viewing directions at the same timeis normally not possible. Therefore, the spectacle lenses aremanufactured such that they cause a good correction of vision disordersof the eye and only small aberrations particularly in the main zones ofuse, especially in the central visual zones, while larger aberrationsare permitted in peripheral zones.

In order to be able to manufacture a spectacle lens in this way, thespectacle lens surfaces or at least one of the spectacle lens surfacesis first calculated such that the desired distribution of theunavoidable aberrations is effected thereby. This calculation andoptimization is usually performed by means of an iterative variationmethod by minimizing a target function. As a target function,particularly a function F having the following functional relationshipto the spherical power S, the magnitude of the cylindrical power Z, andthe axis of the cylinder Δ (also referred to as “SZA” combination) istaken into account and minimized:

$F = {\sum\limits_{i = 1}^{m}\left\lbrack {{g_{i,{S\Delta}}\left( {S_{\Delta,i} - S_{\Delta,i,{target}}} \right)}^{2} + {g_{i,{Z\Delta}}\left( {Z_{\Delta,i} - Z_{\Delta,i,{target}}} \right)}^{2} + \ldots} \right\rbrack}$

In the target function F, at the evaluation points i of the spectaclelens, at least the actual refractive deficits of the spherical powerS_(Δ,i) and the cylindrical power Z_(Δ,i) as well as targetspecifications for the refractive deficits of the spherical powerS_(Δ,i,target) and the cylindrical power Z_(Δ,i,target) are taken intoconsideration.

It was found in DE 103 13 275 that it is advantageous to not indicatethe target specifications as absolute values of the properties to beoptimized, but as their deviation from the prescription, i.e. as therequired local maladjustment. The advantage is that the targetspecifications are independent of the prescription (Sph_(v), Zyl_(v),Axis_(v), Pr_(v), B_(v)) and that the target specifications do not haveto be changed for every individual prescription. Thus, as “actual”values of the properties to be optimized, not absolute values of theseoptical properties are taken into account in the target function, butthe deviations from the prescription. This has the advantage that thetarget values can be specified independent of the prescription and donot have to be changed for every individual prescription.

The respective refractive deficits at the respective evaluation pointsare preferably taken into consideration with weighting factors g_(i.SΔ)and g_(i,ZΔ). Here, the target specifications for the refractivedeficits of the spherical power S_(Δ,i,target) and/or the cylindricalpower Z_(Δ,i,target), particularly together with the weighting factorsg_(i.SΔ) and g_(i,ZΔ), form the so-called spectacle lens design. Inaddition, particularly further residues, especially further parametersto be optimized, such as coma and/or spherical aberration and/or prismand/or magnification and/or anamorphic distortion, etc., can be takeninto consideration, which is particularly implied by the expression “+.. . ” in the above-mentioned formula for the target function F.

In some cases, consideration not only of aberrations up to the secondorder (sphere, magnitude of the astigmatism, and cylinder axis) but alsoof higher order (e.g. coma, trefoil, spherical aberration) may in somecases contribute to a clear improvement particularly of an individualadaptation of a spectacle lens.

It is known from the prior art to determine the shape of a wavefront foroptical elements and particularly spectacle lenses that are delimited byat least two refractive boundary surfaces. For example, this can be doneby means of a numerical calculation of a sufficient number ofneighboring rays, along with a subsequent fit of the wavefront data byZernike polynomials. Another approach is based on local wavefronttracing in the refraction (cf. WO 2008/089999 A1). Here, only one singleray (the main ray) per visual point is calculated, accompanied by thederivatives of the vertex depths of the wavefront according to thetransversal coordinates (perpendicular to the main ray). Thesederivatives can be formed up to a specific order, wherein the secondderivatives describe the local curvature properties of the wavefront(such as refractive power, astigmatism), and the higher derivatives arerelated to the higher-order aberrations.

In the tracing of light through a spectacle lens, the local derivativesof the wavefront are calculated at a suitable position in the beam pathin order to compare them with desired values obtained from therefraction of the spectacle lens wearer. As the position at which thewavefronts are evaluated, usually the vertex sphere or e.g. theprincipal plane of the eye for the corresponding direction of sight isconsidered. In this respect, it is assumed that a spherical wavefrontemanates from the object point and propagates up to the first spectaclelens surface. There, the wavefront is refracted and subsequentlypropagates to the second spectacle lens surface, where it is refractedagain. The last propagation takes place from the second boundary surfaceto the vertex sphere (or the principal plane of the eye), where thewavefront is compared with the predetermined values for the correctionof the refraction of the eye of the spectacle wearer.

To make this comparison on the basis of the determined refraction dataon the respective eye, an established model of the eye having the visiondisorder, in which a base eye having normal vision is overlaid with avision disorder (refractive deficit), is assumed for the evaluation ofthe wavefront at the vertex sphere. This has proven particularlysuccessful as further knowledge of the anatomy or optics of therespective eye (e.g. distribution of the refractive powers, eye length,length ametropia and/or refractive power ametropia) is not required. Adetailed description of this model of spectacle lens and refractivedeficit can be found e.g. in Dr. Roland Enders “Die Optik des Auges undder Sehhilfen”, Optische Fachveröffentlichung GmbH, Heidelberg, 1995,pages 25 ff. und in Diepes, Blendowske “Optik und Technik der Brille”,Optische Fachveröffentlichung GmbH, Heidelberg, 2002, pages 47 ff. As atried and tested model, in particular the described correction modelaccording to REINER is used.

Here, the refractive deficit is considered to be the lack or excess ofrefractive power of the optical system of the eye having the visiondisorder compared with an equally long eye having normal vision(residual eye). The refractive power of the refractive deficit is inparticular approximately equal to the distance point refraction withnegative sign. For a full correction of the vision disorder, thespectacle lens and the refractive deficit together from a telescopicsystem (afocal system). The residual eye (eye having the vision disorderwithout added refractive deficit) is considered to have normal vision.Thus, a spectacle lens is said to be fully correcting for distance ifits image-side focal point coincides with the distance point of the eyehaving the vision disorder and thus also with the object-side focalpoint of the refractive deficit.

Document DE 10 2017 007 975 A1 or WO 2018/138140 A2, to which referenceis expressly made herein or the content of which is fully included inthe present description, describes a method and a device that allowimproving the calculation or optimization of a spectacle lens, thespectacle lens being adapted very effectively to the individualrequirements of the spectacle wearer with simple measurements ofindividual, optical and eye-anatomical data.

Patients with implanted intraocular lenses (IOL), i.e. patients who haveundergone cataract surgery (e.g. as a result of cataract), are currentlyonly provided with commercially available spectacle lenses, which arealso used for patients with natural eye lenses. Thus, there arecurrently no specially adapted lenses for patients with IOLs. This meansthat the specifics of implanted IOLs cannot be taken into account,although the properties of implanted lenses differ significantly fromthose of natural lenses. For example, IOLs have a different sphericalpower for at least partially compensating for a corneal or lengthametropia and/or a different cylindrical power for at least partiallycompensating for a corneal astigmatism or for at least partiallycancelling out a lens astigmatism. In the case of conventionallyoptimized spectacle lenses, model-based properties are assumed for theeye lens, so that possibly the eye model used for the optimization inthese cases no longer corresponds to the actual structure of the eyewith an IOL, e.g. if a length ametropia is at least partiallycompensated for by the IOL due to the power (mean sphere) of the IOL.

SUMMARY

It is the object of the present invention to improve the calculation oroptimization of a spectacle lens, preferably a progressive spectaclelens. In particular, it can be an object to provide patients withspecially designed spectacle lenses after cataract surgery. Inparticular, it can be an object to improve the calculation oroptimization of a spectacle lens, preferably a progressive spectaclelens, with regard to patients having an implanted intraocular lens. Thisobject is achieved in particular by a computer-implemented method, adevice, a computer program product, a storage medium, and acorresponding spectacle lens with the features specified in theindependent claims. Preferred embodiments are the subject of thedependent claims.

A first aspect for solving the object relates to a computer-implementedmethod for identifying relevant individual parameters of at least oneeye of a spectacle wearer for the calculation or optimization of anophthalmic lens (in particular a spectacle lens) for the at least oneeye of the spectacle wearer, comprising the steps of:

-   -   providing individual data on properties of the at least one eye        of the spectacle wearer;    -   constructing an individual eye model by defining (and/or        specifying) a set of parameters of the individual eye model; and    -   determining a probability distribution (or a probability        density) of values of the parameters of the individual eye model        on the basis of the individual data.

The individual data on properties of the at least one eye of thespectacle wearer include, in particular, measurement data on propertiesof the at least one eye of the spectacle wearer. These known individualdata can e.g. include: refraction data (in particular a currentsubjective and/or objective refraction and/or an earlier subjectiveand/or objective refraction, the term earlier referring to a point intime prior to surgery, for example), the power and/or shape and/orposition (in particular the axial position) of certain refractivesurfaces of the eye, the size and/or shape and/or position of theentrance pupil, the refractive index of the refractive media, therefractive index profile in the refractive media, the opacity, etc.

Defining a set of parameters of the individual eye model is understoodin particular to mean that the eye model is set up or defined via or bya specific set of parameters. In other words, it is established whichparameters the eye model is to include and/or by which parameters theindividual eye model is to be defined or characterized. The parametersdefined for the eye model can initially be variables (i.e. parameterswithout a specific value). With the aid of the method described in thecontext of the invention, the parameters or values of the parameters canthen be determined or defined (e.g. as the most likely parameters of theindividual eye model).

In a preferred embodiment, constructing an individual eye modelcomprises providing an initial probability distribution of theparameters of the eye model. Furthermore, determining a probabilitydistribution (or a probability density) of values of the parameters ofthe individual eye model takes place on the basis of the initialprobability distribution of parameters of the eye model. The initialprobability distribution is preferably based on a population analysis.In particular, the initial probability distribution is based on (orcorresponds to) information about the probability distribution of theparameters of the individual eye model in the general public or in thepopulation of certain people.

The method according to the invention is thus based on probabilitycalculation, in which in particular Bayesian statistics can be used for.Correspondingly, in a further preferred embodiment, an individual eyemodel is constructed and/or a probability distribution of values of theparameters of the eye model is determined using Bayesian statistics.

In a further preferred embodiment, determining a probabilitydistribution of values of the parameters of the individual eye modelcomprises calculating a consistency measure, wherein in particular theproduct of the probability or probability density of the individual datafor given parameters of the individual eye model with the probability orprobability density of the parameters of the individual eye model,especially with given background knowledge, is used as the consistencymeasure. The background knowledge (i.e. the current state of knowledgewhen evaluating the data) in particular includes available backgroundinformation, e.g. about the measurement process of the refraction, thedistribution of the parameters of the individual eye model or otherrelated variables in the general public. The probability or probabilitydensity of the parameters of the individual eye model with givenbackground knowledge (also referred to below as prob(

_(i)|I)) can in particular correspond to the “prior” of Bayesianstatistics. The probability or probability density of the individualdata for given parameters of the individual eye model (hereinafter alsoreferred to as prob(d_(i)|

_(i), I) can in particular correspond to the “likelihood” of Bayesianstatistics. Moreover, the consistency measure prob(

_(i)|d_(i), I) proportional to prob(

|I)prob(d_(i)|

_(i), I) can in particular correspond to the “posterior” of Bayesianstatistics.

In a further preferred embodiment, the method further comprises thesteps of:

-   -   calculating a probability distribution (or a probability        density) of parameters of the ophthalmic lens to be calculated        or optimized using at least one parameter of the individual eye        model; and preferably    -   determining most likely values of the parameters of the        ophthalmic lens to be calculated or optimized, in particular by        maximizing the probability for the individual parameters of the        ophthalmic lens to be calculated or optimized.

As explained in more detail below, the probability distribution (or theprobability density) of parameters L_(i) of the ophthalmic lens to becalculated or optimized can in particular be specified as follows:

$\begin{matrix}{{{prob}\left( {\left. L_{i} \middle| d_{i} \right.,I} \right)} = {\int{d\vartheta_{i}{{prob}\left( {\vartheta_{i},\left. L_{i} \middle| d_{i} \right.,I} \right)}}}} \\{= {\int{d\vartheta_{i}{{prob}\left( {\left. L_{i} \middle| \vartheta_{i} \right.,I} \right)}{{prob}\left( {\left. \vartheta_{i} \middle| d_{i} \right.,I} \right)}}}} \\{= {\int{d\vartheta_{i}{\delta\left( {L_{i} - {L\left( \vartheta_{i} \right)}} \right)}{{prob}\left( {\left. \vartheta_{i} \middle| d_{i} \right.,I} \right)}}}}\end{matrix}$

In a further preferred embodiment, providing individual data comprisesproviding individual refraction data on the at least one eye of thespectacle wearer. Furthermore, constricting an individual eye modelcomprises defining an individual eye model in which at least

-   -   a shape and/or power of a cornea, in particular a corneal front        surface (18), of a model eye (12); and/or    -   a cornea-lens distance; and/or    -   parameters of the lens of the model eye; and/or    -   a lens-retina distance; and/or    -   a size of the entrance pupil; and/or    -   a size and/or position of a physical aperture diaphragm (or iris        opening) can be determined in particular on the basis of        individual measurement values for the eye of the spectacle        wearer and/or standard values and/or on the basis of the        provided individual refraction data. The method also includes        the following steps of:    -   carrying out a consistency check of the defined eye model with        the provided individual refraction data, and    -   solving any inconsistencies, in particular with the aid of        analytical and/or numerical and/or probabilistic methods.

If the eye model includes parameters of the cornea, the position andsize of the aperture diaphragm of the eye can be converted into theposition and size of the entrance pupil and vice versa, since theentrance pupil represents the aperture diaphragm imaged by the cornea.In particular, it can therefore be sufficient if in this case theposition or the size of either the aperture diaphragm or of the entrancepupil is used as a (possibly additional) parameter of the eye model.

A computer-implemented method for determining relevant individualparameters of at least one eye of a spectacle wearer for the calculationor optimization of a spectacle lens for the at least one eye of thespectacle wearer, the at least one eye of the spectacle wearer having animplanted intraocular lens, can in particular comprise the followingsteps of:

-   -   providing intraocular lens data on the intraocular lens        implanted in the eye of the spectacle wearer;    -   providing individual refraction data on the at least one eye of        the spectacle wearer; and    -   defining an individual eye model, in which in particular at        least        -   a shape and/or power of a cornea, in particular a corneal            front surface, of a model eye;        -   a cornea-lens distance;        -   parameters of the lens of the model eye;        -   a lens-retina distance;

are defined on the basis of the provided intraocular lens data andfurther on the basis of individual measurement values for the eye of thespectacle wearer and/or standard values and/or on the basis of theprovided individual refraction data such that the model eye has theprovided individual refraction data, wherein defining the parameters ofthe lens of the model eye is carried out on the basis of the providedintraocular lens data. Preferably, at least defining the lens-retinadistance is carried out by calculation.

The intraocular lens can in particular be an aphakic intraocular lens ora phakic intraocular lens. An aphakic intraocular lens replaces thenatural eye lens, i.e. the at least one eye of the spectacle wearer onlyhas the intraocular lens after surgery (but no longer the natural lens).A phakic intraocular lens, in contrast, is inserted or implanted in theat least one eye of the spectacle wearer in addition to the natural eyelens, i.e. the at least one eye of the spectacle wearer has both theintraocular lens and the natural eye lens after surgery.

In the context of the present invention, the term “lens of the modeleye” can refer not only to a single real lens (e.g. the natural eye lensor an intraocular lens), but also to a lens system. The lens system caninclude one or more lenses, in particular two lenses (namely the naturallens of the eye and also an intraocular lens). In other words, in thecontext of the present invention, the “lens of the model eye” can be alens, in particular a model-based or virtual lens, which describes thenatural eye lens or an (aphakic) intraocular lens. Alternatively, in thecontext of the present invention, the “lens of the model eye” can alsobe a in particular model-based or virtual lens system, which describesthe natural eye lens and additionally a (phakic) intraocular lens. Forexample, the “lens of the model eye” can be understood as a thick lensof the model eye (or the “lens of the model eye” can be a thick lens ofthe model eye), which combines and/or describes both the properties ofthe natural eye lens and the properties of an additional intraocularlens. In particular, in the context of the present invention, the term“lens of the model eye” is understood to mean a “lens system of themodel eye”. Correspondingly, in the context of the present invention,the term “lens-retina distance” is understood to mean in particular a“lens system-retina distance”. For the sake of simplicity, however, onlythe terms “lens of the model eye” and “lens-retina distance” will beused in the following.

A phakic intraocular lens can e.g. be considered as an own/separate oradditional element in the eye model. Alternatively, the power of aphakic intraocular lens can influence the power (or one of the surfacesor both surfaces) of the cornea. In this case, the intraocular lens canbe an anterior chamber lens, for example. As a further alternative, thepower of a phakic intraocular lens can influence the power (or one ofthe surfaces or in both surfaces) of the natural eye lens. In this case,the intraocular lens can be a posterior chamber lens, for example. Ifthe intraocular lens is introduced as an additional element, itsproperties (in particular power, surfaces and/or thickness) can be takeninto account in the eye model as additional parameters of the eye model.For example, the position of this additional lens can either be definedas a known position or taken into account as an additional parameter(e.g. measured or model-based) in the eye model. The defined positioncan (in the case of an anterior chamber lens) e.g. be directly or at acertain distance behind the cornea, or e.g. (in the case of a posteriorchamber lens) directly or at a certain distance in front of the eyelens.

The ophthalmic lens or the spectacle lens can be optimized according toone of the methods described in WO 2013/104548°A1 or DE 10 2017 007 974A1 by tracing into the eye. As described in DE 10 2017 007 975 A1 or WO2018/138140 A2, the eye model required for this is assigned individualvalues, with the known properties of the IOL, e.g. from themanufacturer's information, being included in the lens of the eye model.The intraocular lens data includes data on properties of the implantedintraocular lens. These can be specified directly when ordering orobtained from a database by specifying the type or the individual serialnumber. This is helpful or advantageous, since the model-based valuesused in DE 10 2017 007 975 A1 or WO 2018/138140 A2 are not necessarilyconsistent with the properties of the lens actually implanted. This canbe the case, for example, when a length myopia is at least partiallycompensated for by an IOL with less refraction. In this case, the methoddescribed in DE 10 2017 007 975 A1 or WO 2018/138140 A2 would result intoo short an eye length.

The eye model preferably comprises various components such as cornea,lens, retina, etc. and parameters or a set of parameters of thesecomponents. Parameters of the eye model are e.g. the shape of a cornealfront surface of the model eye, the cornea-lens distance, the parametersof the lens of the model eye, the lens-retina distance, etc.

In a preferred embodiment, the distance lens-retina (hereinafter DLR ord_(LR)) is calculated from the defocus term of the entire eye, thedefocus term of the corneal surface, the distance cornea-lens(hereinafter DCL or d_(CL)) and the data on the IOL with one of theformalisms described in DE 10 2017 007 975 A1 or WO 2018/138140 A2. Theterm defocus term will be used below for the value of the symmetricalsecond term (c_(2,0)) of the Zernike decomposition of the refractivepower or the surface of an optical element.

In particular, one or more of the following data can be used:

-   -   IOL data: This can either be the defocus of the refractive        surfaces (front and back surface) and a propagation length        (thickness of the lens, hereinafter DLL) or the defocus of the        refractive power of the IOL. While a model based on the surfaces        and the distance can deliver more precise results during        optimization, optimization requires more calculation steps        (refraction-propagation-refraction instead of just refraction)        and correspondingly detailed information about the IOL, which        may not be available.    -   Defocus term of the entire eye: The result of an aberrometric        measurement or an autorefraction, a subjective refraction or        another determination (e.g. retinoscopy) can be used here.        Alternatively, a so-called “optimized refraction”, i.e. the        result of a calculation from several components (e.g. subjective        refraction and aberrometry) can be used. Examples of such an        optimization are compiled in DE 10 2017 007 975 A1 or WO        2018/138140 A2. In particular, the individual refraction data on        the at least one eye of the spectacle wearer include a defocus        or defocus term of the (overall) eye.    -   Defocus term of the cornea: It can be taken e.g. from a        topography or topometry measurement or it can be assumed based        on a model.    -   Distance cornea-lens: It can be determined with a measurement        (e.g. Scheimpflug imaging or OCT) or assumed based on a model.

Instead of or in addition to the defocus, another variable, preferably asecond-order term, such as the power in the main section with thehighest or lowest refractive power or in a meridian with a definedposition (e.g. horizontal or vertical) can be used.

Depending on requirements, the eye model can be supplemented by theastigmatism (magnitude and axis, or the other second-order variablesaccording to the previous paragraph) as well as higher-order components(see DE 10 2017 007 975 A1 or WO 2018/138140 A2) of the entire eye andthe components. These can be taken e.g. from the IOL data, measurements(e.g. topography/topometry or aberrometry/autorefraction), modelassumptions and/or calculated values (e.g. optimized refraction).

Often, information about asphericity or higher-order aberrations isgiven for IOLs. These can be used when higher-order aberrations areassigned to the eye model.

First of all, individual refraction data on the at least one eye of thespectacle wearer are provided. This individual refraction data is basedon an individual refraction determination. The refraction data includesat least the spherical and astigmatic vision disorder of the eye. In apreferred embodiment, the acquired refraction data also describeshigher-order aberrations (HOA). Preferably, the refraction data (alsoreferred to as aberrometric data in particular if they includehigher-order aberrations) is measured by an optician, for example, usingan autorefractometer or an aberrometer (objective refraction data).Alternatively or in addition, a subjectively determined refraction canbe used as well. Subsequently, the refraction data will preferably becommunicated to a lens manufacturer and/or provided to a calculation oroptimization program. The data is therefore available to be acquired, inparticular to be read out and/or received in digital form for the methodaccording to the invention.

Preferably, providing the individual refraction data comprises providingor identifying the vergence matrix S_(M) of the vision disorder of theat least one eye. Here, the vergence matrix describes a wavefront infront of the eye of the light emanating from a point on the retina orconverging in a point on the retina. In terms of measurement technology,such refraction data can be determined e.g. by illuminating a point onthe retina of the spectacle wearer by means of a laser, from which pointlight then propagates. While the light from the illuminated pointinitially diverges substantially spherically in the vitreous body of theeye, the wavefront can change when passing through the eye, inparticular at optical boundary surfaces in the eye (e.g. the eye lensand/or the cornea). The refraction data on the eye can thus be measuredby measuring the wavefront in front of the eye.

In addition, the method can comprise defining an individual eye model,which individually defines at least certain specifications aboutgeometric and optical properties of a model eye. Thus, in the individualeye model according to the invention, at least a shape (topography)and/or power of the cornea, in particular of a corneal front surface ofthe model eye, a cornea-lens distance d_(CL) (this distance between thecornea and a lens front surface of the model eye is also referred to asanterior chamber depth), parameters of the lens of the model eye, whichin particular at least partially define the optical power of the lens ofthe model eye, and a lens-retina distance d_(LR) (this distance betweenthe lens, in particular the lens back surface, and the retina of themodel eye is also referred to as the vitreous body length) is defined ina specific way, namely in such a way that the model eye has the providedindividual refraction data, i.e. that a wavefront emanating in the modeleye from a point on the retina of the model eye matches with thewavefront identified (e.g. measured or otherwise identified) for thereal eye of the spectacle wearer (up to a desired accuracy). Asparameters of the lens of the model eye (lens parameters), for exampleeither geometric parameters (shape of the lens surfaces and theirdistance) and preferably material parameters (e.g. refractive indices ofthe individual components of the model eye) can be defined so completelythat they at least partially define an optical power of the lens.Alternatively or in addition, parameters directly describing the opticalpower of the lens of the model eye can also be defined as lensparameters. With regard to the cornea, the shape of the corneal frontsurface is usually measured, but alternatively or in addition the powerof the cornea as a whole (no differentiation between front and backsurfaces) can be specified. A corneal back surface and/or a cornealthickness can possibly also be specified as well.

The parameters of the lens of the model eye can be defined (exclusively)on the basis of the provided intraocular lens data. In particular, theparameters of the lens of the model eye correspond to the individualprovided intraocular lens data. In other words, the provided intraocularlens data is defined as the parameters of the lens of the model eye.

In the simplest case of an eye model, the refraction of the eye isdetermined by the optical system consisting of the corneal frontsurface, the eye lens, and the retina. In this simple model, therefraction of light on the corneal front surface and the refractivepower of the eye lens (preferably including the spherical and astigmaticaberrations and higher-order aberrations) together with theirpositioning relative to the retina define the refraction of the modeleye.

Here, the individual variables (parameters) of the model eye are definedaccordingly on the basis of the provided intraocular lens data andfurther on the basis of individual measurement values for the eye of thespectacle wearer and/or on the basis of standard values and/or on thebasis of the provided individual refraction data. In particular, some ofthe parameters (e.g. the topography of the corneal front surface and/orthe anterior chamber depth and/or at least a curvature of a lenssurface, etc.) can be provided directly as individual measurementvalues. Other values can also be taken over from values of standardmodels for a human eye, especially if the parameters involved are verycomplex to measure individually. Overall, however, not all (geometric)parameters of the model eye have to be specified from individualmeasurements or from standard models. Rather, in the context of theinvention, an individual adaptation is carried out for one or more(free) parameters by performing a calculation taking into account thepredefined parameters such that the then-resulting model eye has theprovided individual refraction data. Depending on the number ofparameters included in the provided individual refraction data, acorresponding number of (free) parameters of the eye model can beindividually adapted (fitted). Deviating from a model proposed e.g. inWO°2013/104548°A1, at least the lens-retina distance can be defined bycalculation in the context of the present invention.

For the calculation or optimization of the spectacle lens, a firstsurface and a second surface of the spectacle lens are specified inparticular as starting surfaces with a predetermined (individual)position relative to the model eye. In a preferred embodiment, only oneof the two surfaces is optimized. This is preferably the back surface ofthe spectacle lens. A corresponding starting surface is preferablyspecified for both the front surface and the back surface of thespectacle lens. In a preferred embodiment, however, only one surface isiteratively changed or optimized during the optimization process. Theother surface of the spectacle lens can be a simple spherical orrotationally symmetrical aspherical surface, for example. However, it isalso possible to optimize both surfaces.

Starting from the two predetermined surfaces, the method for calculatingor optimizing comprises determining the path of a main ray through atleast one visual point (i) of at least one surface of the spectacle lensto be calculated or optimized into the model eye. The main ray describesthe geometric beam path emanating from an object point through the twospectacle lens surfaces, the corneal front surface, and the lens of themodel eye preferably up to the retina of the model eye.

In addition, the method for calculating or optimizing can compriseevaluating an aberration of a wavefront resulting from a sphericalwavefront incident on the first surface of the spectacle lens along themain ray on an evaluation surface in particular in front of or withinthe model eye compared to a wavefront (reference light) converging inone point on the retina of the eye model.

In particular, a spherical wavefront (w₀) incident on the first surface(front surface) of the spectacle lens along the main ray is specifiedfor this purpose. This spherical wavefront describes the light emanatingfrom an object point (object light). The curvature of the sphericalwavefront when incident on the first surface of the spectacle lenscorresponds to the reciprocal of the object distance. The method thuspreferably comprises specifying an object distance model that assigns anobject distance to each viewing direction or to each visual point of theat least one surface of the spectacle lens to be optimized. Thispreferably describes the individual wearing situation in which thespectacle lens to be manufactured is to be used.

The wavefront incident on the spectacle lens is now refracted on thefront surface of the spectacle lens preferably for the first time. Thewavefront then propagates along the main ray within the spectacle lensfrom the front surface to the back surface, where it is refracted forthe second time. Preferably, the wavefront transmitted through thespectacle lens now propagates along the main ray up to the corneal frontsurface of the eye, where it is preferably refracted again. Preferably,after further propagation within the eye up to the eye lens, thewavefront is also refracted again there in order to finally propagatepreferably up to the retina of the eye. Depending on the opticalproperties of the individual optical elements (spectacle lens surfaces,corneal front surface, eye lens), each refraction process also leads toa deformation of the wavefront.

In order to achieve an exact mapping of the object point to an imagepoint on the retina, the wavefront would preferably have to leave theeye lens as a converging spherical wavefront, the curvature of whichcorresponds exactly to the reciprocal value of the distance to theretina. A comparison of the wavefront emanating from the object pointwith a wavefront (reference light) converging in a point on the retina(in the ideal case of a perfect image) thus allows the evaluation of amismatch. This comparison and thus the evaluation of the wavefront ofthe object light in the individual eye model can take place at differentpoints along the path of the main ray, in particular between the secondsurface of the optimizing spectacle lens and the retina. In particular,the evaluation surface can thus be at different positions, in particularbetween the second surface of the spectacle lens and the retina. Therefraction and propagation of the light emanating from the object pointis calculated accordingly in the individual eye model, preferably foreach visual point. The evaluation surface can either relate to theactual beam path or to a virtual beam path such as is used to constructthe exit pupil AP, for example. In the case of the virtual beam path,the light must be propagated back through the back surface of the eyelens after refraction up to a desired level (preferably up to the levelof the AP), wherein the refractive index used must correspond to themedium of the vitreous body and not to the eye lens. If the evaluationsurface is provided behind the lens or after the refraction on the lensback surface of the model eye, or if the evaluation surface is reachedby back-propagation along a virtual beam path (as in the case of theAP), then the resulting wavefront of the object light can preferablysimply be compared to a spherical wavefront of the reference light. Tothis end, the method thus preferably comprises specifying a sphericalwavefront incident on the first surface of the spectacle lens,identifying a wavefront resulting from the spherical wavefront due tothe power at least of the first and second surfaces of the spectaclelens, the corneal front surface, and the lens of the model eye in the atleast an eye, and evaluating the aberration of the resulting wavefrontin comparison to a spherical wavefront converging on the retina.

If, however, an evaluation surface is to be provided within the lens orbetween the lens of the model eye and the spectacle lens to becalculated or optimized, a reverse propagation from a point on theretina through the individual components of the model eye up to theevaluation surface is simulated as the reference light, in order to makea comparison of the object light with the reference light there.

However, as already mentioned at the beginning, a complete correction ofthe refraction of the eye is generally not possible simultaneously forall viewing directions of the eye, i.e. for all visual points of the atleast one spectacle lens surface to be optimized. Depending on theviewing direction, intentional maladjustment of the spectacle lens isthus preferably specified, which, depending on the applicationsituation, is small especially in the mainly used zones of the spectaclelens (e.g. central visual points), and somewhat higher in the less-usedzones (e.g. peripheral visual points). In principle, this procedure isalready known from conventional optimization methods.

To optimize the spectacle lens, the at least one surface of thespectacle lens to be calculated or optimized is varied iteratively untilan aberration of the resulting wavefront corresponds to a specifiedtarget aberration, i.e. in particular deviates from the wavefront of thereference light (e.g. a spherical wavefront whose center of curvature ison the retina) by specified values. The wavefront of the reference lightis also referred to as a reference wavefront here. Preferably, themethod comprises minimizing a target function F, in particular analogousto the target function described at the beginning. Further preferredtarget functions, in particular when taking higher-order aberrationsinto account, will be described further below. If a propagation of theobject light up to the retina is calculated, an evaluation can becarried out there instead of a comparison of wavefront parameters, forexample by means of a so-called “point spread function”.

In the context of the present invention, it is therefore suggested, inparticular for the calculation or optimization of a spectacle lens, thatsuch an individual eye model that is individually adapted to theindividual spectacle wearer up to the retina be defined that at leastthe vitreous body length of the model eye is calculated individually asa function of other individually identified, in particular measurementdata on the eye. This parameter does not have to be defined a-prior, nordoes it have to be measured directly.

In the context of the present invention, it was found that this broughtabout a remarkable improvement in the individual adaptation withcomparatively little effort, because the wavefront tracing turned out tobe very sensitively dependent on this length parameter.

The individual calculation of the eye model, in particular thelens-retina distance (vitreous body length), can already be carried oute.g. in an aberrometer or a topograph with a correspondingly expandedfunctionality. Preferably, the length of an eye is identifiedindividually. Particularly preferably, the measured and/or calculatedvitreous body length and/or the identified (measured and/or calculated)eye length is displayed to the user. To this end, a corresponding device(in particular an aberrometer or topograph) has a corresponding displaydevice.

In the context of the present invention, it is particularly suggestedthat known properties of the implanted IOLs be used when calculatingspectacle lenses. This advantageously results in a spectacle lens thatis better adapted for patients with an implanted IOL and has optimizedimaging and design preservation on the retina.

In a preferred embodiment, the individual intraocular lens data compriseat least a defocus of the front surface of the intraocular lens, adefocus of the back surface of the intraocular lens, and a thickness ofthe intraocular lens. Alternatively or in addition, the individualintraocular lens data includes at least a defocus of the refractivepower of the intraocular lens or an optical power of the intraocularlens. The intraocular lens data can therefore either be the defocus ofthe refractive surfaces (front and back surfaces) and a propagationlength (thickness of the lens, hereinafter DLL) or the defocus of therefractive power of the IOL. While a model based on the surfaces and thedistance can deliver more precise results during optimization,optimization requires more calculation steps(refraction-propagation-refraction instead of just refraction) andcorrespondingly detailed information about the IOL, which may not beavailable. Alternatively or in addition, the individual intraocular lensdata can include information, in particular a value, relating to aso-called A constant. The A constant is an individual lens constant, inparticular a type of correction factor that can appear in IOLcalculation formulae with different names. It is also known as the IOLconstant or “surgeon factor”. Each IOL from each manufacturer has adifferent A constant that is specified for each calculation formula.This constant represents the intraocular lens in the various calculationformulae. Since all IOL constants can be converted into one another,there is in principle only one constant (number) that is to characterizea given intraocular lens in the entire available power range, regardlessof form factor, optical material, IOL diameter, etc. By using A or IOLconstants, the effects of individual surgical technology, measurementand surgical equipment used, and individual physiological differences inthe cohort of patients undergoing surgery on the IOL calculation areminimized. The A constant particularly reflects any adaptations in thepower and can be part of the lens prescription or IOL prescription.

In a further preferred embodiment, the individual intraocular lens datais provided on the basis of type or serial number information, inparticular by the manufacturer of the IOL. This information can beindicated e.g. directly when ordering or be obtained from a database.

In a further preferred embodiment, the method further comprises thesteps of:

-   -   carrying out a consistency check of the defined eye model, and    -   solving any inconsistencies, in particular with the aid of        analytical and/or numerical and/or probabilistic methods.

In any case, the procedure according to the invention provides aconsistent model with regard to the defocus (or the other variables usedin the calculation of the eye length). However, the consistency of themodel is no longer ensured already with other second-order variables(e.g. magnitude and direction of the astigmatism). In other words, theeye model can be overdetermined and consequently no longer consistent.On the one hand, this can be due to manufacturing inaccuracies of theIOL and measurement inaccuracies, which can occur e.g in the topographyor topometry, aberrometry or autorefraction and/or the measurement ofthe anterior chamber depth. On the other hand, when subjectiverefraction is used, inconsistencies can in principle arise if thesubjective or optimized refraction does not correspond to the objectiveoptical power of the entire eye. In the context of this description, aconsistent eye model is understood to mean an eye model in which anincident wavefront that corresponds to the aberrations of the entire eyeconverges at a point on the retina. This is synonymous with the factthat the wavefront emanating from a point of light on the retinacorresponds to the aberrations of the entire eye after it has passedthrough the entire eye.

A consistency check can in particular be carried out using probabilisticmethods. In this case, a consistency measure could be given as aprobability. Any inconsistencies could be solved e.g. by determining amaximum of the probability.

Carrying out a consistency check and solving any inconsistencies inparticular improves the calculation or optimization of spectacle lensesintended for a patient with IOLs. However, carrying out a consistencycheck and solving any inconsistencies are also advantageous in the caseof spectacle lenses not specifically intended for a patient with IOLs.The present invention thus generally provides a computer-implementedmethod for identifying relevant individual parameters of at least oneeye of a spectacle wearer for the calculation or optimization of aspectacle lens for the at least one eye of the spectacle wearer,comprising the steps of:

-   -   providing individual refraction data on the at least one eye of        the spectacle wearer;    -   defining an individual eye model, in which in particular one or        more of the following information or parameters, namely        -   a shape and/or power of a cornea, in particular a corneal            front surface (18), of a model eye (12); and/or        -   a cornea-lens distance; and/or        -   parameters of the lens of the model eye; and/or        -   a lens-retina distance; and/or        -   a size of the entrance pupil; and/or    -   a size and/or position of a physical aperture diaphragm are        defined on the basis of individual measurement values for the        eye of the spectacle wearer and/or standard values and/or on the        basis of the provided individual refraction data,    -   carrying out a consistency check of the defined eye model, in        particular with the provided individual refraction data, and        optionally    -   solving any inconsistencies, in particular with the aid of        analytical and/or numerical and/or probabilistic procedures or        methods.

“Defining an individual eye model” can mean defining model parameters tobe specific values. Additionally or alternatively, however, “defining anindividual eye model” can also comprise defining at least oneconsistency measure (or at least one probability). In particular, aplurality of values of the model parameters can exist. A consistencymeasure or probability can be defined for each combination of thesevalues. For example, such consistency measures or probabilities can bedefined using Bayes' method.

Preferably, at least the defining of the lens-retina distance is carriedout by calculation. The term “calculation” in the context of the presentinvention can include not only the calculation using an equation, butalso steps that are carried out in a statistical method, such as theselection of values on the basis of statistical considerations orprobabilities. With the Bayes' method, it is possible, for example, thatonly a likely or most likely lens-retina distance is selected or definedby an optimization problem (which then still has to be solved). The term“calculation” in the context of the present invention can thus inparticular also include the selection of likely or most likely values ofone or more parameters and/or the definition of an optimization problem.In particular, the term “calculation” also includes a selection,determination and/or definition in the context of a statisticalprocedure, e.g. in the context of or using the Bayes' method. The term“calculation” can in particular also include optimization.

In addition or as an alternative to performing a consistency check onthe defined eye model and solving any inconsistencies, thecomputer-implemented method can also comprise defining or constructing aconsistent eye model, in particular using Bayes' method and/or a maximumlikelihood method. In other words, the individual eye model used or tobe defined is a consistent eye model, with the consistency being madepossible or established by statistical or probabilistic methods, inparticular using Bayes' method and/or a maximum likelihood method.

In particular, in this aspect there is provided a computer-implementedmethod and a corresponding device for performing such a method foridentifying relevant individual parameters of at least one eye of aspectacle wearer for the calculation or optimization of a spectacle lensfor the at least one eye of the spectacle wearer, comprising one or moreof the following steps or functions:

-   -   providing individual refraction data on the at least one eye of        the spectacle wearer; and/or    -   defining an individual eye model, in which in particular one or        more of the following information or parameters, namely        -   a shape and/or power of a cornea, in particular a corneal            front surface, of a model eye; and/or        -   a cornea-lens distance; and/or        -   parameters of the lens of the model eye; and/or        -   a lens-retina distance; and/or        -   a size of the entrance pupil; and/or        -   a size and/or position of a physical aperture diaphragm

are defined in particular on the basis of individual measurement valuesfor the eye of the spectacle wearer and/or standard values and/or on thebasis of the provided individual refraction data,

wherein one or more of the information or parameters and/or at leastpartially the provided individual refraction data is or are initiallydefined in the form of a probability distribution, and wherein definingthe individual eye model comprises identifying the model eye byidentifying values for information or parameters within the definedprobability distribution by a probabilistic method.

While in some aspects a model eye is first created by defining parametervalues in order to then possibly modify the model eye on the basis of aconsistency check using a probabilistic method so that the eye model isconsistent, in the present case, instead of possibly inconsistentparameter values, a probability distribution is started for at least oneparameter in order to then consistently identify the most likelyparameter value and thus the most probably model eye using aprobabilistic method. Parameters of the probability distribution(s),such as mean values and/or standard deviations, can in particular bedetermined on the basis of individual measurement values for the eye ofthe spectacle wearer and/or standard values and/or on the basis of theprovided individual refraction data. Further details and specificexemplary embodiments of such methods will be described below.

In the following, a few examples are used to describe how anyinconsistencies in the eye model can be eliminated by adapting theparameters with the aid of analytical calculations.

The simplest possibility is to transfer the deviations to an element ora component of the eye model (e.g. cornea, front surface of the IOL,back surface of the IOL, refractive power of the IOL). For example, theback surface of the IOL could (contrary to the manufacturer'sinstructions) be chosen so that the model is consistent. For thispurpose (when the defocus term is used), after the calculation of DLR,first the astigmatism, in particular according to magnitude anddirection (e.g. according to the method described inDE°10°2017°007°975°A1 or WO°2018/138140°A2) can be defined so that theeye model becomes consistent in terms of astigmatism. Furthermore,higher-order components of this surface (e.g. with the help of themethod described in DE°10°2017°007°975°A1 or WO°2018/138140°A2) can bedefined in subsequent steps, for example, so that the eye model is alsoconsistent in these components. Alternatively or in addition, thecorneal surface could be adapted accordingly. This is particularlyuseful if only model-based information on the cornea or no informationon astigmatism or higher-order components are available due totopometric measurements.

In a further preferred embodiment, any inconsistencies are solved byadapting or redefining one or more parameters of the eye model.Preferably, several parameters of the eye model are adapted and theadaptation is divided among the plurality of parameters of the eyemodel. For example, known deviations can be divided among severalelements or components and/or several parameters of the eye model, e.g.the cornea, the front surface of the IOL, the back surface of the IOL,and/or the refractive power of the IOL. In the simplest case, fixed orpredetermined factors or proportions can be assumed, e.g. 33% on thecornea and 67% on the lens. Alternatively or in addition, aphysiologically based distribution can be used as well.

Alternatively or in addition, a further or new parameter can be added tothe eye model and defined such that the eye model becomes consistent.For example, the shape of the corneal back surface of a model eye can besuch a further parameter. In the case of toric lenses with fixedastigmatism, for example, the cylinder axis and/or a lateral shift ortilt can be selected so that the resulting astigmatism of the model eyecorresponds to the specification (as best as possible).

Alternatively or in addition, the lengths DCL, DLL and/or DLR can beadapted. If necessary, the power of the entire eye can also be adjusted.Here, the target power of the spectacle lens can be changed accordinglyin order to render the eye model consistent.

In a further preferred embodiment, the parameters of the eye model aredetermined with the aid of probabilistic methods, i.e. using probabilitycalculations. For this purpose, in particular Bayesian statistics and/ora maximum likelihood algorithm can be used.

Instead of or in addition to the analytical calculation of the eyelength on the basis of a set of parameters, in particular all knownparameters (hereinafter input parameters) can be combined and theparameters of the eye model (hereinafter output parameters) can bedetermined with the aid of statistical methods such as maximumlikelihood and Bayes. Here, one or more of the following information onat least individual input parameters can be used:

-   -   confidence in the correctness;    -   measurement and manufacturing accuracy;    -   fluctuation range in a collective or ensemble;    -   effect on the optimization of the spectacle lens.

A description of two such methods and specific examples will be givenbelow in the detailed description.

In a further preferred embodiment, an initial distribution of parametersof the eye model and individual data on properties of the at least oneeye are provided, the parameters of the individual eye model beingdetermined on the basis of the initial distribution of parameters of theeye model and the individual data using probability calculations. Inother words, an initial eye model and individual data on properties ofthe at least one eye are provided, the parameters of the individual eyemodel being determined on the basis of the initial eye model and theindividual data using probability calculations.

In a further preferred embodiment, an eye length of the model eye isdetermined taking into account the measured and/or calculatedlens-retina distance. Preferably, the identified eye length is displayedon a display device or display.

The method described above relates in particular to the case thatproperties or data on an implanted intraocular lens, i.e. theintraocular lens data, are known. However, if this data is not known, analternative approach is proposed in the context of this invention, whichwill be described below. According to this alternative approach of thepresent invention, i.e. if there is no direct knowledge of theproperties of the implanted lens, conclusions are drawn regarding theproperties of the implanted IOLs by measurements on the patient.

An alternative approach to solving the object (namely in the event thatproperties or data on the implanted intraocular lens are not known)relates to a computer-implemented method for identifying relevantindividual parameters of at least one eye of a spectacle wearer for thecalculation or optimization of a spectacle lens for the at least one eyeof the spectacle wearer, with an intraocular lens having been implantedin the at least one eye of the spectacle wearer as part of surgery,comprising the steps of:

-   -   providing individual post-surgery refraction data on the at        least one eye of the spectacle wearer;    -   identifying a lens-retina distance (or an eye length) of the eye        of the spectacle wearer; and    -   defining an individual post-surgery eye model, in which in        particular at least        -   a shape and/or power of a cornea, in particular a corneal            front surface, of a model eye of the post-surgery eye model;        -   a cornea-lens distance of the model eye of the post-surgery            eye model;        -   parameters of the lens of the model eye of the post-surgery            eye model; and        -   a lens-retina distance of the model eye of the post-surgery            eye model; are defined on the basis of the identified            lens-retina distance (or the eye length) and further on the            basis of individual measurement values for the eye of the            spectacle wearer (identified prior to or after surgery)            and/or standard values and/or on the basis of the provided            individual post-surgery refraction data so that the model            eye of the post-surgery eye model has the provided            individual post-surgery refraction data, with the            lens-retina distance of the model eye of the post-surgery            eye model being defined by the identified lens-retina            distance of the eye of the spectacle wearer.

In particular, in a further aspect there can be provided acomputer-implemented method for identifying relevant individualparameters of at least one eye of a spectacle wearer for the calculationor optimization of a spectacle lens for the at least one eye of thespectacle wearer, with an intraocular lens having been implanted in theat least one eye of the spectacle wearer as part of surgery, the methodin particular comprising the following steps of:

-   -   providing individual refraction data on the at least one eye of        the spectacle wearer; and    -   defining an individual eye model, in which at least        -   a shape and/or power of a cornea, in particular a corneal            front surface, of a model eye;        -   a cornea-lens distance;        -   parameters of the lens of the model eye; and        -   a lens-retina distance;

are defined as parameters of the individual eye model, wherein definingthe parameters of the individual eye model takes place on the basis ofdata on visual acuity correction of the at least one eye having theintraocular lens and further on the basis of individual measurementvalues for the eye of the spectacle wearer and/or standard values and/oron the basis of the provided individual refraction data such that themodel eye has the provided individual refraction data.

Accordingly, in a further aspect there can be provided a device fordetermining relevant individual parameters of at least one eye of aspectacle wearer for the calculation or optimization of a spectacle lensfor the at least one eye of the spectacle wearer, the at least one eyeof the spectacle wearer having an implanted intraocular lens, the devicein particular comprising:

-   -   at least one data interface for providing individual refraction        data on the at least one eye of the spectacle wearer; and    -   a modeling module for defining an individual eye model, which at        least defines        -   a shape and/or power of a cornea, in particular a corneal            front surface (18), of a model eye;        -   a cornea-lens distance;        -   parameters of the lens of the model eye; and        -   a lens-retina distance;

as parameters of the individual eye model, wherein defining theparameters of the individual eye model takes place on the basis of dataon visual acuity correction of the at least one eye having theintraocular lens and further on the basis of individual measurementvalues for the eye of the spectacle wearer and/or standard values and/oron the basis of the provided individual refraction data such that themodel eye has the provided individual refraction data.

In a preferred embodiment, the data for the visual acuity correction ofthe at least one eye having the intraocular lens include (in particularindividual) intraocular lens data. Thus, in this embodiment, theparameters of the individual eye model are defined on the basis ofintraocular lens data and further on the basis of individual measurementvalues for the eye of the spectacle wearer and/or standard values and/oron the basis of the provided individual refraction data so that themodel eye has the provided individual refraction data, with theparameters of the lens of the model eye being defined on the basis ofthe intraocular lens data.

In a further preferred embodiment, a lens-retina distance of the eye ofthe spectacle wearer is identified, and the parameters of the individualeye model are defined on the basis of the identified lens-retinadistance and further on the basis of individual measurement values forthe eye of the spectacle wearer and/or standard values and/or on thebasis of the provided individual refraction data such that the model eyehas the provided individual refraction data, with the lens-retinadistance of the model eye being defined by the identified lens-retinadistance of the eye of the spectacle wearer. In other words, in thisembodiment, the data for the visual acuity correction of the at leastone eye having the intraocular lens includes an identified lens-retinadistance. In particular, the data for the visual acuity correction ofthe at least one eye having the intraocular lens can include anidentified lens-retina distance and/or intraocular lens data.

In particular, in a further aspect there is provided acomputer-implemented method (and a corresponding device) for identifyingrelevant individual parameters of at least one eye of a spectacle wearerfor the calculation or optimization of a spectacle lens for the at leastone eye of the spectacle wearer, with an intraocular lens having beenimplanted in the at least one eye of the spectacle wearer as part ofsurgery or the at least one eye of the spectacle wearer (in particularinstead of or in addition to the natural eye lens) having an implantedintraocular lens. The method can in particular comprise the followingsteps of:

-   -   providing individual refraction data on the at least one eye of        the spectacle wearer; and    -   defining an individual eye model, in which at least        -   a shape and/or power of a cornea, in particular a corneal            front surface, of a model eye;        -   a cornea-lens distance;        -   parameters of the lens of the model eye; and        -   a lens-retina distance;

are defined as parameters of the individual eye model, wherein:

a) defining the parameters of the individual eye model takes place onthe basis of intraocular lens data and further on the basis ofindividual measurement values for the eye of the spectacle wearer and/orstandard values and/or on the basis of the provided individualrefraction data such that the model eye has the provided individualrefraction data, wherein the parameters of the lens of the model eye aredefined on the basis of the intraocular lens data; and/or

b) a lens-retina distance of the eye of the spectacle wearer isidentified and defining the parameters of the individual eye model takesplace on the basis of the identified lens-retina distance and further onthe basis of individual measurement values for the eye of the spectaclewearer and/or standard values and/or on the basis of the providedindividual refraction data such that the model eye has the providedindividual refraction data, wherein the lens-retina distance of themodel eye is defined by the identified lens-retina distance of the eyeof the spectacle wearer.

The term surgery (German: Operation) is generally abbreviated as OP. Theterm “post-surgery” (German: Nach-OP) refers to a situation aftersurgery, while the term “pre-surgery” (German: Vor-OP) refers to asituation before surgery. For example, the surgery is a cataract surgeryin which the natural eye lens is replaced by an intraocular lens.However, it can also be a surgery on an aphakic eye (eye without a lens)in which an intraocular lens is inserted or implanted in the patient'seye. The intraocular lens can therefore in particular represent areplacement for the natural eye lens. In particular, the natural lens ofthe wearer's eye has been replaced by an intraocular lens duringsurgery.

The spectacle lens is preferably optimized according to one of themethods described in WO°2013/104548°A1 or DE°10°2017° 007°974°A1 bytracing into the eye. By analogy with the description in DE 10 2017 007975 A1 or WO 2018/138140 A2, the eye model required for this is assignedindividual values. In this case, however, no information about the IOLis available and model-based values are not necessarily consistent withthe actually implanted lens. This can be the case, for example, if alength myopia is at least partially compensated for by an IOL with lessrefraction. In this case, the eye length would be assumed to be tooshort according to the procedure described in DE 10 2017 007 975 A1 orWO 2018/138140 A2. Therefore, based on data that corresponds to asituation in which the original lens was located in the eye, the eyelength or a lens-retina distance, as described in DE 10 2017 007 975 A1or WO 2018/138140 A2, will be calculated.

Subsequently, the other parameters (i.e. the parameters of the eye lens,in this case the implanted IOL) will be determined on the basis of thethus-calculated eye length (or the calculated lens-retina distance) andthe post-surgery values for the aberrations of the entire eye, thesurface of the cornea, and the distance cornea-lens such that the powerof this eye model corresponds to the aberrations of the entire eye. Thisdiffers fundamentally from the procedure in DE 10 2017 007 975 A1 or WO2018/138140 A2 in that all values of the lens (in this case theimplanted IOL) are determined and that not as in DE 10 2017 007 975 A1or WO 2018/138140 A2 a second-order term (e.g. defocus) is alreadyknown. After determination of this term, however, further terms can bedetermined as described in DE 10 2017 007 975 A1 or WO 2018/138140 A2.These can be further second-order terms and possibly (e.g. in furthersteps) higher-order terms.

For the calculation of the eye length or lens-retina distance, thefollowing data is preferably used specifically:

-   -   defocus of the entire eye before replacement of the lens: This        can be the result of an aberrometric measurement or an        autorefraction, a subjective refraction or another determination        (e.g. retinoscopy) before the surgical procedure. Alternatively,        a so-called “optimized refraction”, i.e. the result of a        calculation from several components (e.g. subjective refraction        and aberrometry) can be used. Examples of such an optimization        are compiled in DE 10 2017 007 975 A1 or WO 2018/138140 A2.        Furthermore, the defocus of the refractive power of older        spectacles worn before the surgical procedure can be used;    -   model-based values for the refractive power or the structure of        the eye lens;    -   measured or model-based values for the cornea-lens distance; and    -   measured or model-based values for the defocus of the cornea.

The data on the last two points can be either from measurements beforethe surgical procedure (operation) or after the surgical procedure. Theuse of data determined after the surgical procedure is particularlyuseful if no corresponding measurements have been carried out before thesurgical procedure.

By analogy, the following data is preferably used to calculateproperties of the lens:

-   -   aberrations of the entire eye after replacement of the lens:        These can be the result of an aberrometric measurement or an        autorefraction, a subjective refraction or another determination        (e.g. retinoscopy) after the surgical procedure. Alternatively,        a so-called “optimized refraction”, i.e. the result of a        calculation from several components (e.g. subjective refraction        and aberrometry) can be used. Examples of such an optimization        are compiled in DE 10 2017 007 975 A1 or WO 2018/138140 A2.    -   the previously determined distance lens-retina;    -   measured or model-based values for the distance cornea-lens; and    -   measured or model-based values for the aberrations of the        cornea.

The data on the last two points can be either from measurements beforethe surgical procedure (operation) or after the surgical procedure. Theuse of data determined before the surgical procedure is particularlyuseful if no corresponding measurements have been carried out after thesurgical procedure.

Providing the individual intraocular lens data can in particularcomprise the following steps of:

-   -   identifying an eye length on the basis of data corresponding to        a situation in which the original natural lens was still in the        eye of the spectacle wearer (situation before the intraocular        lens was implanted);    -   calculating the individual intraocular lens data on the basis of        the identified eye length, the provided individual refraction        data, measured or model-based values for the cornea-lens        distance, and measured or model-based values for aberrations of        the cornea.

The following table exemplarily includes three scenarios. It isunderstood, however, that other combinations are also part of theinvention.

Values for second step (Substitution of the Values for first stepparameters that are still (Calculation of DCL) open) Scenario Data to betransmitted 1 Only data (aberrometry, topography, Only data(aberrometry, topography, possibly distance cornea-lens, possiblydistance cornea-lens, possibly subjective refraction) from subjectiverefraction) from before surgery (e.g. from an order) after surgery (e.g.from an order) Reference number of the order before surgery, data(aberrometry, topography, possibly distance cornea-lens, subjectiverefraction) from after surgery (e.g. from an order) 2 Only subjectiverefraction/values of Only data (aberrometry, the spectacles from beforesurgery topography, possibly distance Topography and, if necessary,cornea-lens, subjective distance cornea-lens before surgery refraction)from after surgery (e.g. from an order) Subjective refraction/values ofthe spectacles from before surgery as well as data (aberrometry,topography, possibly distance cornea-lens, subjective refraction) fromafter surgery (e.g. from an order) 3 Only data (aberrometry, topography,Only subjective refraction from possibly distance cornea-lens, aftersurgery, further data possibly subjective refraction) from (topography,possibly distance before surgery (e.g. from an order) cornea-lens) frombefore surgery Reference number of the order before surgery, subjectiverefraction from after surgery

The determination of a lens-retina distance or an eye length of the eyeof the spectacle wearer can take place by direct measurement, forexample.

In a preferred embodiment, the method further comprises providingindividual pre-surgery refraction data on the at least one eye of thespectacle wearer, wherein the determination of a lens-retina distance oran eye length of the eye of the spectacle wearer on the basis of anindividual pre-surgery eye model takes place using the providedindividual pre-surgery refraction data.

In a preferred embodiment, in the pre-surgery eye model, particularly atleast

-   -   a shape and/or power of a cornea, in particular a corneal front        surface, of a model eye of the pre-surgery eye model;    -   a cornea-lens distance of the model eye of the pre-surgery eye        model;    -   parameters of the lens of the model eye of the pre-surgery eye        model; and    -   a lens-retina distance of the model eye of the pre-surgery eye        model are defined on the basis of individual measurement values        for the eye of the spectacle wearer (determined before or after        surgery) and/or standard values and/or on the basis of the        provided individual pre-surgery refraction data such that the        model eye has the provided individual pre-surgery refraction        data, wherein at least defining the lens-retina distance        preferably takes place by measuring and/or calculating.

The corneal front surface is preferably measured individually and theeye lens of the individual pre-surgery eye model is calculatedaccordingly in order to meet the individually determined pre-surgeryrefraction data. Here, in a preferred embodiment, the corneal frontsurface (or its curvature) is measured individually along the mainsections (topometry). In a further preferred embodiment, the topographyof the corneal front surface (i.e. the complete description of thesurface) is measured individually. In a further preferred embodiment,the cornea-lens distance is defined on the basis of individualmeasurement values for the cornea-lens distance.

Particularly preferably, defining the parameters of the lens of thepre-surgery model eye comprises defining the following parameters:

-   -   a shape of the lens front surface;    -   a lens thickness; and    -   a shape of the lens back surface.

Even if it is not essential for the use of the invention, it is possibleto further improve the individual adaptation using this more precisemodel of the lens.

In this case, in a particularly preferred embodiment, defining the lensthickness and the shape of the lens back surface takes place on thebasis of predetermined values (standard values, for example from thetechnical literature), wherein defining the shape of the lens frontsurface further preferably comprises:

-   -   providing standard values for a mean curvature of the lens front        surface; and    -   calculating the shape of the lens front surface taking into        account the provided individual refraction data.

In a further preferred embodiment of the more detailed lens model,defining the shape of the lens front surface comprises:

-   -   providing an individual measurement value of a curvature in a        normal section of the lens front surface.

In this case, it is particularly preferred that defining the lensthickness and the shape of the lens back surface take place on the basisof standard values, and even more preferably defining the shape of thelens front surface comprises:

-   -   calculating the shape of the lens front surface taking into        account the provided individual refraction data and the provided        individual measurement value of the curvature in a normal        section of the lens front surface.

As an alternative or in addition to the shape of the lens or the lenssurfaces, defining the lens parameters can include defining an opticalpower of the lens. In particular, at least one position of at least onemain plane and a spherical power (or at least one focal length) of thelens of the model eye are defined. A cylindrical power (magnitude andaxial position) of the lens of the model eye is also particularlypreferred. In a further preferred embodiment, optical higher-orderaberrations of the lens of the model eye can also be identified.

Another independent aspect for solving the object relates to acomputer-implemented method for calculating or optimizing an ophthalmiclens (in particular a spectacle lens) for at least one eye of aspectacle wearer, comprising:

-   -   a method for identifying relevant individual parameters of the        at least one eye of the spectacle wearer according to the        present invention;    -   specifying a first surface and a second surface for the        spectacle lens to be calculated or optimized;    -   identifying the course of a main ray through at least one visual        point of at least one surface of the spectacle lens to be        calculated or optimized into the model eye;    -   evaluating an aberration of a wavefront resulting from a        spherical wavefront incident on the first surface of the        spectacle lens along the main ray on an evaluation surface        compared to a wavefront converging in one point on the retina of        the eye model;    -   iteratively varying the at least one surface of the spectacle        lens to be calculated or optimized until the evaluated        aberration corresponds to a predetermined target aberration.

Another independent aspect for solving the object relates to acomputer-implemented method for calculating or optimizing an ophthalmiclens (in particular a spectacle lens) for at least one eye of aspectacle wearer, comprising:

-   -   a method according to the invention for identifying relevant        individual parameters of the at least one eye of the spectacle        wearer;    -   specifying a first surface and a second surface for the        spectacle lens to be calculated or optimized;    -   identifying the course of a main ray through at least one visual        point of at least one surface of the spectacle lens to be        calculated or optimized into the model eye;    -   evaluating an aberration of a wavefront resulting from a        spherical wavefront incident on the first surface of the        spectacle lens along the main ray on an evaluation surface        compared to a wavefront converging in one point on the retina of        the eye model;    -   iteratively varying the at least one surface of the spectacle        lens to be calculated or optimized until the evaluated        aberration corresponds to a predetermined target aberration.

Preferably, the evaluation surface is located between the corneal frontsurface and the retina. In a particularly preferred embodiment, theevaluation surface is located between the lens and the retina of themodel eye. In another particularly preferred embodiment, the evaluationsurface is located on the exit pupil (AP) of the model eye. Here, theexit pupil can be located in front of the lens back surface of the modeleye. With this positioning, a particularly precise, individualadaptation of the spectacle lens can be achieved.

Another independent aspect for solving the object relates to a methodfor producing an ophthalmic lens (in particular a spectacle lens),comprising:

-   -   calculating or optimizing a spectacle lens according to the        inventive method for calculating or optimizing a spectacle lens;        and    -   manufacturing the thus-calculated or optimized spectacle lens.

Another independent aspect for solving the object relates to a devicefor identifying relevant individual parameters of at least one eye of aspectacle wearer for the calculation or optimization of an ophthalmiclens for the at least one eye of the spectacle wearer, comprising:

-   -   at least one data interface for providing individual data on        properties of the at least one eye of the spectacle wearer; and    -   a modeling module for modeling and/or constructing an individual        eye model by defining (and/or specifying) a set of parameters of        the individual eye model; wherein the modeling module is        configured to determine a probability distribution of values of        the parameters of the individual eye model on the basis of the        individual data, in particular using probability calculation.

In a preferred embodiment, providing individual data comprises providingindividual refraction data on the at least one eye of the spectaclewearer. Furthermore, constructing an individual eye model comprisesdefining an individual eye model in which at least

-   -   a shape and/or power of a cornea, in particular a corneal front        surface (18), of a model eye (12); and/or    -   a cornea-lens distance; and/or    -   parameters of the lens of the model eye; and/or    -   a lens-retina distance; and/or    -   a size of the entrance pupil; and/or    -   a size and/or position of a physical aperture diaphragm

are defined in particular on the basis of individual measurement valuesfor the eye of the spectacle wearer and/or standard values and/or on thebasis of the provided individual refraction data. The modeling module isalso configured to carry out a consistency check of the defined eyemodel with the provided individual refraction data and to solve anyinconsistencies, in particular with the aid of analytical and/orprobabilistic methods.

In particular, the invention thus offers a device for identifyingrelevant individual parameters of at least one eye of a spectacle wearerfor the calculation or optimization of an ophthalmic lens (in particulara spectacle lens) for the at least one eye of the spectacle wearer,comprising:

-   -   at least one data interface for providing individual data on        properties of the at least one eye of the spectacle wearer; and    -   a modeling module for defining an individual eye model, which in        particular at least defines        -   a shape and/or power of a cornea, in particular a corneal            front surface (18), of a model eye (12); and/or        -   a cornea-lens distance; and/or        -   parameters of the lens of the model eye; and/or        -   a lens-retina distance; and/or        -   a size of the entrance pupil; and/or        -   a size and/or position of a physical aperture diaphragm

on the basis of individual measurement values for the eye of thespectacle wearer and/or standard values and/or on the basis of theprovided individual refraction data, wherein at least defining thelens-retina distance preferably takes place by measuring and/orcalculating; and wherein

the modeling module is configured to carry out a consistency check ofthe defined eye model with the provided individual refraction data andto solve any inconsistencies, in particular with the aid of analyticaland/or probabilistic methods.

In particular, the invention provides a device for identifying relevantindividual parameters of at least one eye of a spectacle wearer for thecalculation or optimization of a spectacle lens for the at least one eyeof the spectacle wearer, with the at least one eye of the spectaclewearer having an implanted intraocular lens (in particular instead of orin addition to the natural eye lens), comprising:

-   -   at least one data interface for providing individual intraocular        lens data on the intraocular lens implanted in the eye of the        spectacle wearer and for providing individual refraction data on        the at least one eye of the spectacle wearer; and    -   a modeling module for defining an individual eye model, which in        particular at least defines        -   a shape and/or power of a cornea, in particular a corneal            front surface, of a model eye;        -   a cornea-lens distance;        -   parameters of the lens of the model eye; and        -   a lens-retina distance

on the basis of the individual provided intraocular lens data andfurther on the basis of individual measurement values for the eye of thespectacle wearer and/or standard values and/or on the basis of theprovided individual refraction data such that the model eye has theprovided individual refraction data, wherein defining the parameters ofthe lens of the model eye is carried out on the basis of the providedintraocular lens data. Preferably, defining the lens-retina distance iscarried out by measuring and/or calculating.

Preferably, the modeling module is configured to identify an eye lengthof the model eye taking into account the measured and/or calculatedlens-retina distance. The device preferably also comprises a displaydevice for displaying the measured and/or calculated lens-retinadistance and/or the determined eye length. The device is particularlypreferably designed as an aberrometer and/or as a topograph.

Preferably, the modeling module is configured to carry out a consistencycheck of the identified eye model, in particular the identifiedpre-surgery eye model and/or the identified post-surgery eye model.Furthermore, the modeling module is preferably configured to solve anyinconsistencies, in particular with the aid of analytical and/orprobabilistic methods (probability calculation, e.g. using Bayesianstatistics and/or a maximum likelihood approach).

In particular, the invention provides a device for identifying relevantindividual parameters of at least one eye of a spectacle wearer for thecalculation or optimization of a spectacle lens for the at least one eyeof the spectacle wearer, with an intraocular lens being implanted in theat least one eye of the spectacle wearer during surgery, comprising:

-   -   at least one data interface for providing individual        post-surgery refraction data on the at least one eye of the        spectacle wearer; and    -   a modeling module for identifying a lens-retina distance of the        eye of the spectacle wearer and for defining an individual        post-surgery eye model, which in particular at least defines        -   a shape and/or power of a cornea, in particular a corneal            front surface, of a model eye of the post-surgery eye model;        -   a cornea-lens distance of the model eye of the post-surgery            eye model;        -   parameters of the lens of the model eye of the post-surgery            eye model; and        -   a lens-retina distance of the model eye of the post-surgery            eye model

on the basis of the determined lens-retina distance and further on thebasis of individual measurement values for the eye of the spectaclewearer (identified before or after surgery) and/or standard valuesand/or on the basis of the provided individual post-surgery refractiondata such that the model eye of the post-surgery eye model has theprovided individual post-surgery refraction data, wherein thelens-retina distance of the model eye of the post-surgery eye model isdefined by the identified lens-retina distance of the spectacle wearer'seye.

Another independent aspect for solving the object relates to a devicefor calculating or optimizing a spectacle lens for at least one eye of aspectacle wearer, comprising:

-   -   a device according to the invention for identifying relevant        individual parameters of the at least one eye of the spectacle        wearer;    -   a surface model database for specifying a first surface and a        second surface for the spectacle lens to be calculated or        optimized;    -   a main ray identification module for identifying the course of a        main ray through at least one visual point of at least one        surface of the spectacle lens to be calculated or optimized into        the model eye;    -   an evaluation module for evaluating an aberration of a wavefront        resulting from a spherical wavefront incident on the first        surface of the spectacle lens along the main ray on an        evaluation surface compared to a wavefront converging in one        point on the retina of the eye model; and    -   an optimization module for iteratively varying the at least one        surface of the spectacle lens to be calculated or optimized        until the evaluated aberration corresponds to a predetermined        target aberration.

Another independent aspect for solving the object relates to a devicefor producing a spectacle lens, comprising:

-   -   calculation or optimization means configured to calculate or        optimize the spectacle lens according to an inventive method for        calculating or optimizing a spectacle lens; and    -   machining means configured to machine the spectacle lens in        accordance with the result of the calculation or optimization.

The device for producing a spectacle lens can be designed in one pieceor as an independent machine, i.e. all components of the device (inparticular the calculation or optimization means and the machiningmeans) can be part of one and the same system or one and the samemachine. In a preferred embodiment, however, the device for producing aspectacle lens is not designed in one piece, but is realized bydifferent (in particular independent) systems or machines. For example,the calculation or optimization means can be realized as a first system(in particular comprising a computer) and the machining means as asecond system (in particular a machine comprising the machining means).Here, the different systems can be located in different places, i.e.they can be locally separated from one another. For example, one or moresystems can be located in the front end and one or more other systems inthe back end. The individual systems can e.g. be located at differentcompany locations or operated by different companies. The individualsystems in particular have communication means in order to exchange datawith one another (for example via a data carrier). Preferably, thevarious systems of the device can communicate with one another directly,in particular via a network (e.g. via a local network and/or via theInternet). The above statements regarding the device for producing aspectacle lens do not only apply to this device, but also generally toall of the devices described in the context of the present invention. Inparticular, a device described herein can be designed as a system. Thesystem can in particular comprise several devices (possibly locallyseparated) configured to carry out individual method steps of acorresponding method.

In addition, the invention offers a computer program product or acomputer program article, in particular in the form of a storage mediumor a data stream containing program code that is designed, when loadedand executed on a computer, to execute a method according to theinvention for identifying relevant individual parameters of at least oneeye of a spectacle wearer and/or to execute a method according to theinvention for calculating or optimizing a spectacle lens. In particular,a computer program product is to be understood as a program stored on adata carrier. In particular, the program code is stored on a datacarrier. In other words, the computer program product comprisescomputer-readable instructions that, when loaded into a memory of acomputer and executed by the computer, cause the computer to execute amethod according to the invention.

Furthermore, the invention provides a spectacle lens produced by amethod according to the invention and/or using a device according to theinvention.

In addition, the invention provides a use of a spectacle lens producedby the production method according to the present invention, inparticular in a preferred embodiment, in a predetermined average orindividual wearing position of the spectacle lens in front of the eyesof a specific spectacle wearer for correcting a vision disorder of thespectacle wearer.

The invention can in particular comprise one or more of the followingaspects:

-   -   the spectacle lens as a product;    -   the calculation and manufacture of the spectacle lens;    -   the calculation of properties (in particular of designs and        surfaces) of the spectacle lens;    -   the calculation of the eye length and the assignment to an eye        model (also for purposes other than lens calculation);    -   a method, a device and/or a system for acquiring the relevant        data, e.g. in the form of or as part of ordering and/or industry        software, in particular for manual input and/or for importing        from measuring devices and/or databases;    -   a method, a device and/or a system, in particular a protocol,        for the transmission of the relevant data;    -   a method, a device and/or a system for storing the relevant        data, which can be different from the method, the device and/or        the system for calculating the spectacle lens;    -   a method, a device and/or a system for providing and retrieving        the data on the IOLs, in particular on the basis of type or        serial number information by the manufacturer of the IOL, the        calculator of the spectacle lens, or a third party;    -   devices and computer program products for implementing the above        points.

In particular, a computer-implemented method according to the inventioncan be provided in the form of ordering and/or industry software. Inparticular, the data required for the calculation and/or optimizationand/or manufacture of a spectacle lens, in particular the intraocularlens data and/or the prescription data and/or the individual refractiondata (pre-surgery and/or post-surgery refraction data) of the at leastone eye of the spectacle wearer, can be acquired and/or transmitted. Theintraocular lens data can be transmitted e.g. from the manufacturer ofthe intraocular lens data to the calculator and/or manufacturer of thespectacle lens. The prescription data and/or individual refraction datacan be transmitted e.g. from the optician and/or ophthalmologist orsurgeon to the calculator and/or manufacturer of the spectacle lens.Alternatively or in addition, it may be possible to retrieve this datafrom a database, in particular with the help of a type and/or serialnumber of the implanted IOL or with the help of a patient code (e.g.customer or patient number, name, etc.). Measurement or refraction datacan also be called up directly from a measuring device, for example. Acommon transmission protocol or a transmission protocol speciallydeveloped for the method according to the invention can be used for thetransmission of the data. As an alternative or in addition, the data tobe transmitted can also be input, at least in part, manually via aninput unit. In this way, an ophthalmologist or surgeon can e.g. transmitthe so-called A constant or IOL constant of the intraocular lens used.In particular, a lens or IOL prescription can also be created semi- orfully automatically on the basis of the transmitted data.

A device according to the invention and/or a system according to theinvention, e.g. for ordering a spectacle lens, can in particularcomprise a computer and/or data server configured to communicate via anetwork (e.g. Internet). The computer is in particular configured toexecute a computer-implemented method, e.g. ordering software forordering at least one spectacle lens, and/or transmission software fortransmitting relevant data (in particular intraocular lens data and/orprescription data and/or refraction data), and/or identificationsoftware for identifying relevant individual parameters of at least oneeye of a spectacle wearer, and/or calculation or optimization softwarefor calculating and/or optimizing a spectacle lens to be produced,according to the present invention.

The statements made above or below regarding the embodiments of thefirst aspect also apply to the above-mentioned further independentaspects or approaches and in particular to preferred embodiments in thisregard. In particular, the statements made above and below on theembodiments of the respective other independent aspects also apply to anindependent aspect of the present invention and to preferred embodimentsin this regard.

BRIEF DESCRIPTION OF THE DRAWINGS

Preferred embodiments and examples of the invention will be explainedbelow by way of example, at least in part with reference to theaccompanying drawings, which show:

FIG. 1 a schematic representation of the physiological and physicalmodel of a spectacle lens and an eye together with a beam path in apredetermined wearing position;

FIG. 2 a graph with an exemplary dependency of P^(mess) on S_(IOL) andL_(2,IOL) to illustrate and explain a method for identifying parametersunder constraint conditions according to a preferred embodiment of thepresent invention;

FIGS. 3 a-3 f marginal prior probability densities as a contourrepresentation of a sample from the prior distribution with equidistantlines of the same probability density for the first example for themethod according to Bayes A;

FIGS. 4 a-4 f marginal posterior probability densities as a contourrepresentation of a sample from the posterior distribution withequidistant lines of the same probability density for the first examplefor the method according to Bayes A;

FIGS. 5 a-5 c histograms of marginal probability densities of M, J₀, andJ₄₅, which represent a sample from the posterior distribution of thepower of the ophthalmic lens, for the first example for the methodaccording to Bayes A or Bayes B;

FIGS. 6 a-6 e marginal prior probability densities as scatter diagramsof a sample from the prior distribution for the second example for themethod according to Bayes A;

FIGS. 7 a-7 e marginal posterior probability densities as scatterdiagrams of a sample from the posterior distribution for the secondexample for the method according to Bayes A;

FIGS. 8 a-8 c histograms of marginal probability densities of M, J₀, andJ₄₅, which represent a sample from the posterior distribution of thepower of the ophthalmic lens, for the second example for the methodaccording to Bayes A or Bayes B; and

FIGS. 9 a-9 c histograms of marginal probability densities of M, J₀, andJ₄₅, which result from a prior distribution, for the second example forthe method according to Bayes A or Bayes B.

DETAILED DESCRIPTION

FIG. 1 shows a schematic representation of the physiological andphysical model of a spectacle lens and an eye in a predetermined wearingposition together with an exemplary beam path on which an individualspectacle lens calculation or optimization according to a preferredembodiment of the invention is based.

Here, only a single ray is preferably calculated for each visual pointof the spectacle lens (the main ray 10, which preferably runs throughthe ocular center of rotation Z′), but also the derivatives of thevertex depths of the wavefront according to the transverse coordinates(perpendicular to the main ray). These derivatives are taken intoaccount up to the desired order, the second derivatives describing thelocal curvature properties of the wavefront and the higher derivativesbeing related to the higher-order aberrations.

Upon light tracing through the spectacle lens up to the eye 12 accordingto the individually provided eye model, the local derivatives of thewavefronts are ultimately identified at a suitable position in the beampath in order to compare them there with a reference wavefront thatconverges in a point on the retina of the eye 12. In particular, the twowavefronts (i.e. the wavefront coming from the spectacle lens and thereference wavefront) are compared with one another in an evaluationsurface.

“Position” does not simply mean a certain value of the z-coordinate (inthe direction of light), but such a coordinate value in combination withthe specification of all surfaces through which refraction took placebefore reaching the evaluation surface. In a preferred embodiment,refraction takes place through all refractive surfaces including thelens back surface. In this case, the reference wavefront is preferably aspherical wavefront whose center of curvature is located on the retinaof the eye 12.

It is particularly preferred not to propagate any further after thislast refraction, so that the radius of curvature of this referencewavefront does correspond to the distance between the lens back surfaceand the retina. In a further preferred embodiment, propagation is stillcarried out after the last refraction, preferably up to the exit pupilAP of eye 12. This is, for example, at a distance d_(AR)=d_(LR)^((b))=d_(LR)−D_(LR) ^((a))>d_(LR) in front of the retina and thus evenin front of the lens back surface, so that the propagation in this caseis a back propagation (the designations d_(LR) ^((a)), d_(LR) ^((b))will be described below in the listing of steps 1-6). In this case, too,the reference wavefront is spherical with the center of curvature on theretina, but has a radius of curvature 1/d_(AR).

To this end, it is assumed that a spherical wavefront w₀ emanates fromthe object point and propagates up to the first spectacle lens surface14. There it is refracted and then it propagates to the second lenssurface 16, where it is refracted again. The wavefront w_(g1) exitingthe spectacle lens then propagates along the main ray in the directionof the eye 12 (propagated wavefront w_(g2)) until it hits the cornea 18,where it is refracted again (wavefront w_(c)). After further propagationwithin the anterior chamber of the eye up to the eye lens 20, thewavefront is also refracted again by the eye lens 20, whereby theresulting wavefront we arises on the back surface of the eye lens 20 oron the exit pupil of the eye, for example. This is compared with thespherical reference wavefront w_(s) and the deviations are evaluated forall visual points in the target function (preferably with correspondingweightings for the individual visual points).

Thus, the vision disorder is no longer described only by a thinsphero-cylindrical lens, as was customary in many conventional methods,but rather the corneal topography, the eye lens, the distances in theeye, and the deformation of the wavefront (including the low-orderaberrations—i.e. sphere, cylinder and cylinder axis—as well aspreferably including the higher-order aberrations) are directly takeninto account in the eye. Here, the vitreous body length d_(LR) iscalculated individually in the eye model according to the invention.

An aberrometer measurement preferably provides the individual wavefrontdeformations of the real eye having the visual defect for distance andnear (deviations, no absolute refractive indices) and the individualmesopic and photopic pupil diameters. From a measurement of the cornealtopography (surface measurement of the corneal front surface), anindividual real corneal front surface is preferably obtained, whichgenerally makes up almost 75% of the total refractive power of the eye.In a preferred embodiment, it is not necessary to measure the cornealback surface. It is preferably described by an adaptation of therefractive index of the cornea and not by a separate refractive surfacedue to the small refractive index difference compared to the aqueoushumor and because of the small cornea thickness in a good approximation.

In general, in this description, bold lowercase letters are intended todenote vectors and bold capital letters are intended to denote matrices,such as the (2×2) vergence matrices or refractive index matrices

${S = \begin{pmatrix}S_{xx} & S_{xy} \\S_{xy} & S_{yy}\end{pmatrix}},{C = \begin{pmatrix}C_{xx} & C_{xy} \\C_{xy} & C_{yy}\end{pmatrix}},{L = \begin{pmatrix}L_{xx} & L_{xy} \\L_{xy} & L_{yy}\end{pmatrix}},{1 = \begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix}}$

and italics like dare intended to denote scalar quantities.

Furthermore, bold, italicized capital letters are intended to denotewavefronts or surfaces as a whole. For example, S is the vergence matrixof the wavefront S of the same name, except that S besides the 2nd orderaberrations summarized in S, also comprises the entirety of allhigher-order aberrations (HOA) of the wavefront. From a mathematicalpoint of view, S stands for the set of all parameters necessary todescribe a wavefront (with sufficient accuracy) in relation to a givencoordinate system. Preferably, S stands for a set of Zernikecoefficients with a pupil radius or a set of coefficients of a Taylorseries. Particularly preferably, S stands for the set of a vergencematrix S for describing the wavefront properties of the 2nd order and aset of Zernike coefficients (with a pupil radius), which is used todescribe all remaining wavefront properties except the 2nd order, or aset of coefficients according to a Taylor decomposition. Analogousstatements apply to surfaces instead of wavefronts.

Among other things, the following data can be measured directly inprinciple:

-   -   The wavefront S_(M), which is generated by the laser spot on the        retina and the passage through the eye (from aberrometric        measurement)    -   Shape of the corneal front surface C (through corneal        topography)    -   Distance between cornea and lens front surface d_(CL) (by        pachymetry). This variable can also be determined indirectly by        measuring the distance between the cornea and the iris;        correction values can be applied if necessary here. Such        corrections can be the distance between the lens front surface        and the iris from known eye models (e.g. literature values).    -   Curvature of the lens front surface in a direction L_(1xx) (by        pachymetry) In this case, without restricting generality, the        x-plane can be defined exemplarily such that this section lies        in the x-plane. If the coordinate system is defined such that        this plane is inclined, the derivative must be supplemented by        the functions of the corresponding angle. It is not required        that this be a main section. For example, it can be the section        in the horizontal plane.

Furthermore, the following data—depending on the embodiment—can eitherbe measured or taken from the literature:

-   -   Thickness of the lens d_(L)    -   Curvature of the lens back surface in the same direction as the        lens front surface L_(2,xx) (by pachymetry)

Thus, there are the following options for the lens back surface:

-   -   Measurement of L_(2,xx) (L_(2,M)) and assumption of a rotational        symmetry L_(2,xx)=L_(2,yy)=L₂=L_(2,M) and L_(2,xy)=L_(2,yx)=0    -   Taking L_(2,xx) from the literature (L_(2,Lit)) and assumption        of a rotational symmetry L_(2,xx)=L_(2,yy)=L₂=L_(2,M) und        L_(2,xy)=L_(2,yx)=0    -   Taking the complete (asymmetrical) form L₂ from the literature        (L_(2,Lit))    -   Measurement of L_(2,xx) (L_(2,M)) and assumption of a cylinder        or an otherwise specified asymmetry a_(Lit) from the literature        L_(2,xx)=L_(2,M) und L_(2,xy)=L_(2,yx)=f (L_(2,xx), a_(Lit)) and        L_(2,yy)=g(L_(2,xx), a_(Lit))

The following data can be found in the literature:

-   -   Refractive indices n_(CL) of the cornea and anterior chamber of        the eye as well as the aqueous humor n_(LR) and that of the lens        n_(L)

This leaves in particular the distance d_(LR) between the lens backsurface and the retina and the components L_(1,yy) and L_(1,xy)=L_(1,yx)of the lens front surface as unknown parameters. To simplify theformalism, the former can also be written as a vergence matrixD_(LR)=D_(LR)·1 with D_(LR)=n_(LR)/d_(LR). Furthermore, the variable τis generally used, which is defined as τ=d/n (whereby for the refractiveindex, always the corresponding index must be used as n, as for d and τ,e.g. as τ_(LR)=d_(LR)/n_(LR), τ_(CL)=d_(CL)/n_(CL)).

The modeling of the passage of the wavefront through the eye model usedaccording to the invention, i.e. after passage through the surfaces ofthe spectacle lens, can be described as follows in a preferredembodiment in which the lens is described via a front and a backsurface, with the transformations the vergence matrices is explicitlybeing specified:

-   -   1. Refraction of the wavefront S with the vergence matrix S on        the cornea C with the surface power matrix C to the wavefront        S′_(C) with vergence matrix S′_(C)=S+C 2. Propagation around the        anterior chamber depth d_(CL) (distance between cornea and lens        front surface) to wavefront S_(L1) with vergence matrix        S_(L1)=S′_(C)/(1−τ_(CL)·S′)

$S_{L1} = \frac{S_{C}^{\prime}}{\left( {1 - {\tau_{CL} \cdot S_{C}^{\prime}}} \right)}$

-   -   3. Refraction on the lens front surface L₁ with the surface        power matrix L₁ to the wavefront S′_(L1) with the vergence        matrix S′_(L1)=S_(L1)+L₁    -   4. Propagation around the lens thickness d_(L) to the wavefront        S_(L2) with vergence matrix S_(L2)=S′_(L1)/(1−τ_(L)·S′_(L1))    -   5. Refraction on the lens back surface L₂ with the surface power        matrix L₂ to the wavefront S′_(L2) with vergence matrix        S′_(L2)=S_(L2)+L₂    -   6. Propagation around the distance between lens and retina        d_(LR) to wavefront S_(R) with the vergence matrix        SS_(R)=S′_(L2)/(1−τ_(LR)·S′_(L2))

Each of the steps 2, 4, 6, in which the distances τ_(CL), τ_(CL), andτ_(CL) are propagated, can be divided into two partial propagations 2a,b), 4a, b) or 6a, b) according to the following scheme, which explicitlyreads for step 6a, b):

-   -   6a. Propagation around the distance d_(LR) ^((a)) between the        lens and the intermediate plane to the wavefront S_(LR) with the        vergence matrix S_(LR)=S′_(L2)/1−τ_(LR) ^((a))S′_(L2))    -   6b. Propagation around the distance d_(LR) ^((b)) between the        intermediate plane and the retina to the wavefront S_(R) with        the vergence matrix S_(R)=S_(LR)/(1−τ_(LR) ^((b))S_(LR))

Here, τ_(LR) ^((a))=d_(LR) ^((a))/n_(LR) ^((a)) and τ_(LR) ^((b))=d_(LR)^((b))/n_(LR) ^((b)) can be positive or negative, whereby it shouldalways hold that n_(LR) ^((a))=n_(LR) ^((b))=n_(LR) and τ_(LR)^((a))+τ_(LR) ^((b))=τ_(LR). In any case, steps 6a and 6b can becombined again by S_(R)=S′_(LR)/(1−(τ_(LR) ^((a))+τ_(LR)^((b)))S′_(L2))=S′_(L2)/(1−τ_(LR)S′_(L2)). The division in steps 6a and6b offers advantages, however, and the intermediate plane can preferablybe placed in the plane of the exit pupil AP, which preferably is locatedin front of the lens back surface. In this case τ_(LR) ^((a))<0 andτ_(LR) ^((b))>0.

The division of steps 2, 4 can be analogous to the division of step 6 in6a, b).

Decisive for the choice of the evaluation surface of the wavefront isnot only the absolute position in relation to the z-coordinate (in thedirection of light), but also the number of surfaces through whichrefraction took place up to the evaluation surface. Thus, one and thesame level can be passed several times. For example, the plane of the AP(which normally is located between the lens front surface and the lensback surface) is formally passed by the light for the first time afteran imaginary step 4a, in which propagation takes place from the lensfront surface by the length τ_(L) ^((a))>0. The same plane is reachedfor the second time after step 6a when, after the refraction through thelens back surface, propagation back to the AP plane takes place, i.e.τ_(LR) ^((a))=−τ_(L)+τ_(L) ^((a))=−τ_(L) ^((b)), which is synonymouswith τ_(LR) ^((a))=−τ_(LR)−τ_(LR) ^((b))<0. In the case of thewavefronts S_(AP) that refer to the AP in the text, the wavefrontS_(AP)=S_(LR) that is the result of step 6a should preferably always bemeant (unless explicitly stated otherwise).

Reference will be made to these steps 1 to 6 again in the further courseof the description. They describe a preferred relationship between thevergence matrix S of a wavefront S on the cornea and the vergencematrices of all intermediate wavefronts resulting therefrom at therefractive intermediate surfaces of the eye, in particular the vergencematrix S′_(L2) of a wavefront S′_(L2) after the eye lens (or even awavefront S_(R) on the retina). These relationships can be used both tocalculate parameters that are not known a priori (e.g. d_(LR) or L₁) andthus to assign values to the model either individually or generically,and to simulate the propagation of the wavefront in the eye foroptimizing spectacle lenses with the models that underwent assignment.

In a preferred embodiment, the surfaces and wavefronts are treated up tothe second order, for which a representation by means of vergencematrices is sufficient. Another preferred embodiment described latertakes into account and also uses higher orders of aberrations.

In a description in the second order, the eye model, in a preferredembodiment, has twelve parameters as degrees of freedom of the modelthat have to undergo assignment. These preferably include the threedegrees of freedom of the surface power matrix C of the cornea C, thethree degrees of freedom of the surface power matrices L₁ and L₂ for thefront and back surfaces of the lens, as well as respectively one for thelength parameters anterior chamber depth d_(CL), lens thickness d_(L),and the vitreous body length d_(LR).

In principle, these parameters can undergo assignment in several ways:

-   -   i. Directly, i.e. individual measurement of a parameter    -   ii. Value of a parameter given a priori, e.g. as a literature        value or from an estimate, for example due to the presence of a        measurement value for another variable, which correlates in a        known manner with the parameter to be determined on the basis of        a previous population analysis    -   iii. Calculation from consistency conditions, e.g. compatibility        with a known refraction

The total number df₂ of degrees of freedom of the eye model in thesecond order (df stands for ‘degree of freedom’, index ‘2’ for secondorder) is thus composed of

df₂=df₂(i)+df₂(ii)+df₂(iii)

For example, if there are direct measurement values for all twelve modelparameters, then df₂(i)=12, df₂(ii)=0 and df₂(iii)=0, which will beexpressed by the notation df₂=12+0+0 in the following for the sake ofsimplicity. In such a case, the objective refraction of the relevant eyeis also defined, so that an objective refraction determination no longerhas to be carried out in addition.

A central aspect of the invention relates precisely to the airm of nothaving to measure all parameters directly. In particular, it issignificantly easier to measure the refraction of the relevant eye or todetermine it objectively and/or subjectively than to measure allparameters of the model eye individually. Preferably, there is thus atleast one refraction, i.e. measurement data for the wavefront S_(M) ofthe eye up to the second order, which corresponds to the data on thevergence matrix S_(M). If the eye model undergoes assignment on thebasis of purely objectively measurement data, these values can be takenfrom aberrometric measurements or autorefractometric measurements or,according to (ii), be assigned with other given data. Consideration ofsubjective methods (i.e. subjective refraction), be it as a replacementfor the objective measurement of the refraction or by combining bothresults, will be described later. The three conditions of conformancewith the three independent parameters of the vergence matrix S_(M) thusmake it possible to derive three parameters of the eye model, whichcorresponds to df₂(iii)=3 in the notation introduced above.

The invention therefore makes use of the possibility, in cases in whichnot all model parameters are accessible to direct measurements or inwhich these measurements would be very complex, to assign the missingparameters in a useful way. For example, if direct measurement values(df₂(i)≤9) are available for a maximum of nine model parameters, thenthe refraction conditions mentioned can be used to calculate three ofthe model parameters (df₂(iii)=3). If exactly df₂(i)=9, then all twelvemodel parameters are uniquely determined by the measurements and thecalculation, and it holds that (df₂ (ii)=0). If, on the other hand, df₂(i)<9, then df₂ (ii)=9−df₂ (i)>0, i.e. the model is underdetermined inthe sense that df₂(ii) parameters must be defined a priori.

With the provision of an individual refraction, i.e. measurement data onthe wavefront S_(M) of the eye, in particular up to the second order,the necessary data on the vergence matrix S_(M) are correspondinglyavailable. According to a conventional method described inWO°2013/104548°A1, in particular the parameters {C, d_(CL), S_(M)} aremeasured. In contrast, the two length parameters d_(L) and d_(LR) (orD_(LR)) are conventionally defined a priori (e.g. by literature valuesor estimates). In WO°2013/104548°A1, a distinction is in particular madebetween the two cases in which either L₂ is defined a priori and L₁ iscalculated therefrom, or vice versa. The aforementioned laid-openpublication discloses Eq. (4) and Eq. (5) as a calculation rule. Forboth cases, it holds that df₂=4+5+3.

In the terminology used in steps 1 to 6 above, L₁ is adapted to themeasurements in particular by calculating the measured vergence matrixS_(M) by means of steps 1, 2 through the likewise measured matrix C andpropagating it up to the object-side side of the lens front surface. Onthe other hand, a spherical wave is calculated back to front from animaginary point-like light source on the retina by means of steps 6, 5,4 carried out backward by refracting this spherical wave at thepreviously defined surface power matrix L₂ of the lens back surface andpropagating the then-obtained wavefront from the lens back surface up tothe image-side side of the lens front surface. The difference of thethus-determined vergence matrices S_(L1) and S′_(L1), which are locatedon the object side or on the image side of the lens front surface, musthave been caused by the matrix L₁, since in the aberrometric measurementthe measured wavefront emerges from a wavefront that emanates from apoint pm the retina, and therefore, due to the reversibility of the beampaths, is identical to the incident wavefront (S=S_(M)) that convergeson this point of the retina. This leads to equation (4) in the laid-openpublication mentioned:

$\begin{matrix}{{L_{1}\left( D_{LR} \right)} = {\frac{{D_{LR} \cdot 1} - L_{2}}{1 + {\tau_{L} \cdot \left( {{D_{LR} \cdot 1} - L_{2}} \right)}} - \frac{S_{M} + C}{1 - {\tau_{CL}\left( {S_{M} + C} \right)}}}} & \left( {1a} \right)\end{matrix}$

The other case in the cited laid-open publication relates to theadaptation of the matrix L₂ to the measurements after the matrix L₁ hasbeen defined. The only difference now is that the measured wavefrontS_(M) is subjected to steps 1, 2, 3, 4 and the assumed wavefront fromthe point-like light source only to step 6, and the missing step to becarried out to adapt the lens back surface L_(z) is now step 5,according to equation (5) of the above-mentioned laid-open publication:

$\begin{matrix}{L_{2} = {D_{LR} - {\left( {\frac{S_{M} + C}{1 - {\tau_{CL}\left( {S_{M} + C} \right)}} + L_{1}} \right)\left( {1 - {\tau_{L}\left( {\frac{S_{M} + C}{1 - {\tau_{CL}\left( {S_{M} + C} \right)}} + L_{1}} \right)}} \right)^{- 1}}}} & \left( {1b} \right)\end{matrix}$

The central idea of the invention is to calculate at least the lengthparameter d_(LR) (or D_(LR)) from other measurement data and a prioriassumptions about other degrees of freedom and not to assume it a prioriitself, as is conventional. In the context of the present invention, itturned out that this brought about a remarkable improvement in theindividual adaptation with comparatively little effort since thewavefront tracing turned out to be very sensitive to this lengthparameter. This means that it is an advantage according to the inventionif at least the length parameter d_(LR) belongs to the df₂(iii)=3parameters that are calculated. In particular, it is difficult tomeasure this parameter directly, it varies more strongly betweendifferent test subjects, and these variations have a comparativelystrong influence on the imaging of the eye.

The data on the vergence matrix S_(M) and particularly preferably alsothe data on C are preferably available from individual measurements. Ina further preferred aspect, which is preferably also taken into accountin the following embodiments, in the assumption of data on the lens backsurface, a spherical back surface, i.e. a back surface withoutastigmatic components is taken as a basis.

In a preferred embodiment of the invention, measurement data up to thesecond order are available for the cornea C, which corresponds to thedata on the surface power matrix C. Although these values can be takenfrom topographical measurements, the latter are not necessary. Rather,topometric measurements are sufficient. This situation corresponds tothe case df₂=3+6+3, with the anterior chamber depth d_(CL) in particularbeing one of the six parameters to be defined a priori.

If no further individual measurements are made, the situation isdf₂=3+6+3. In order to be able to uniquely determine d_(LR), sixparameters from {L₁, L₂, d_(L), d_(CL)} have to undergo assignment byassumptions or literature values. The other two result in addition tod_(LR) from the calculation. In a preferred embodiment, the parametersof the lens back surface, the mean curvature of the lens front surface,and the two length parameters d_(L) and d_(CL) undergo assignment apriori (as predetermined standard values).

In a case that is particularly important for the invention, the anteriorchamber depth d_(CL), i.e. the distance between the cornea and the lensfront surface, is also known e.g. from pachymetric or OCT measurements.The measured parameters thus include {C, d_(CL), S_(M)}. This situationcorresponds to the case df₂=4+5+3. The problem is therefore stillunderdetermined mathematically, so five parameters from {L₁, L₂, d_(L)}have to be deformed a priori through assumptions or literature values.In a preferred embodiment, these are the parameters of the lens backsurface, the mean curvature of the lens front surface, and the lensthickness.

For the accuracy of the individual adaptation alone, it is advantageousto be able to assign individual measurements to as many parameters aspossible. In a preferred embodiment, the lens curvature is additionallyprovided in a normal section on the basis of an individual measurement.This then results in a situation according to df₂=5+4+3, and it issufficient to define four parameters from {L_(1yy), α_(L1), L₂, d_(L)} apriori. Here, too, in a preferred embodiment, the parameters of the lensback surface and the lens thickness are involved. The exact calculationwill be described below.

In particular, as an alternative to the normal section of the lens frontsurface and particularly preferably in addition to the anterior chamberdepth, the lens thickness can also be made available from an individualmeasurement. This eliminates the need to assign model data or estimatedparameters to this parameter (df₂=5+4+3). Otherwise, what has alreadybeen said above applies. This embodiment is particularly advantageous ifa pachymeter is used, the measuring depth of which allows the lens backsurface to be recognized, but not a sufficiently reliable determinationof the lens curvatures.

In addition to the anterior chamber depth and a normal section of thelens front surface, one (e.g. measurement in two normal sections) or twofurther parameters (measurement of both main sections and the axialposition) of the lens front surface can be obtained by an individualmeasurement. This additional information can be exploited in two ways inparticular:

-   -   Abandonment of a priori assumptions: One or two of the        assumptions otherwise made a priori can be abandoned and        determined by calculation. In this case the situations df₂=6+3+3        and df₂=7+2+3 arise. In the first case, the mean curvature of        the back surface (assuming an astigmatism-free back surface) and        in the second case the surface astigmatism (including cylinder        axis) with given mean curvature can be determined.        Alternatively, the lens thickness can also be determined from        the measurements in both cases.    -   However, such a procedure generally requires a certain degree of        caution, since noisy measurement data can easily lead to the        released parameters “running away”. As a result, the model can        become significantly worse instead of better overall. One        possibility to prevent this is to specify anatomically sensible        limit values for these parameters and to limit the variation of        the parameters to this range. Of course, these limits can also        be specified as a function of the measurement values.    -   Reduction of the measurement uncertainty: If, on the other hand,        the same a priori assumptions are made (preferably {L₂, d_(L)}),        the situations df₂=6+4+3 and df₂=7+4+3 exist, the system is        therefore mathematically overdetermined.    -   Instead of a simple analytical determination of D_(LR) according        to the following explanations, D_(LR) (and possibly the missing        parameter from L₁) is determined (“fit”) such that the distance        between L₁ resulting from the equations and the measured L₁ (or        L₁ supplemented by the missing parameter) becomes minimal. By        this procedure—obviously—a reduction in the measurement        uncertainty can be achieved.

In a further preferred implementation, the anterior chamber depth, twoor three parameters of the lens front surface, and the lens thicknessare measured individually. The other variables are calculated in thesame way, whereby the a priori assumption of the lens thickness can bereplaced by the corresponding measurement.

In a further preferred implementation, individual measurements of theanterior chamber depth, at least one parameter of the lens frontsurface, the lens thickness, and at least one parameter of the lens backsurface are provided. This is an addition to the cases mentioned above.The respective additionally measured parameters can be carried outanalogously to the step-by-step expansions of the above sections. Thesecases are particularly advantageous if the above-mentioned pachymetryunits, which measure in one plane, two planes or over the entiresurface, are correspondingly expanded in the measuring depth and are soprecise that the curvature data can be identified with sufficientaccuracy.

If the formula (1b) already mentioned above is solved for D_(LR) inorder to calculate the lens back surface for a given eye length (whereD_(LR)=n_(LR)/d_(LR) is the inverse vitreous body length d_(LR)multiplied by the refractive index n_(LR)), namely

$L_{2} = {D_{LR} - {\left( {\frac{S_{M} + C}{1 - {\tau_{CL}\left( {S_{M} + C} \right)}} + L_{1}} \right)\left( {1 - {\tau_{L}\left( {\frac{S_{M} + C}{1 - {\tau_{CL}\left( {S_{M} + C} \right)}} + L_{1}} \right)}} \right)^{- 1}}}$

one obtains

$D_{LR} = {L_{2} + {\left( {\frac{S_{M} + C}{1 - {\tau_{CL}\left( {S_{M} + C} \right)}} + L_{2}} \right) \times \left( {1 - {\tau_{L}\left( {\frac{S_{M} + C}{1 - {\tau_{CL}\left( {S_{M} + C} \right)}} + L_{2}} \right)}} \right)^{- 1}}}$

for calculating D_(LR). Since D_(LR) is a scalar, all quantities for thecalculation must also be taken as scalar. Preferably, S_(M), C, L₁ andL₂ are each the spherical equivalents of the vision disorder, thecornea, the lens front surface, and the lens back surface, respectively.Once the vitreous body length d_(LR) has been calculated (and thus theeye length as d_(A)=d_(C)+d_(CL)+d_(L)+d_(LR)), one of the surfaces canbe modified again with regard to the cylinder and the HOA, preferablythe lens front surface L1, in order to adapt it consistently.

For the calculation, the values of the so-called Bennett & Rabbetts eyefor the refractive powers of the lens surfaces can be used, which can betaken from Table 12.1 of the book “Bennett & Rabbets' Clinical VisualOptics”, third edition, by Ronald B. Rabbetts, Butterworth-Heinemann,1998, ISBN-10: 0750618175, for example. The calculation described aboveleads to results that are very compatible with the populationstatistics, which state that short-sighted vision disorders tend to leadto large eye lengths and vice versa (see e.g. C. W. Oyster: “The HumanEye”, 1998). The calculation described is, however, even more precise,since a direct use of the correlation from the population statistics canlead to unphysical values for the eye lens, which is avoided by themethod according to the invention. Precise knowledge of the lensparameters is all the more important for a method in which these areassumed to be given. For example, if a customer had a vision disorder of−10 dpt before his cataract surgery, he must have an eye length between28 mm and 30 mm. After surgery, however, due to his emmetropia, an eyelength of 24 mm would be concluded, which does not match the actual eyelength.

Preferred embodiments of the invention will be explained below.

Examples Using Bayesian Statistics

The aim of the method using Bayesian statistics is to use, if possible,all available information sources about an eye or a pair of eyes in aconsistent manner in order to achieve an optimal correction of the eyeor the eyes with an ophthalmic lens (e.g. a spectacle lens) in the lightof this information.

As a rule, this information is incomplete and/or imprecise, which so farhas often led to the fact that only simplified eye models are used tocalculate ophthalmic lenses. Such a simplified eye model is e.g. an eyethat is characterized solely by its refraction, since this can bedetermined with a certain accuracy (e.g. with an error of ±0.75 dpt inthe spherical equivalent). However, if one wishes to use more complexeye models to calculate ophthalmic lenses, it makes sense to includeinformation about the length of the eye, as well as the position andcurvature of the refractive surfaces of the cornea and eye lens in thecalculation, but this should only be taken into account as much as ispossible within the scope of its accuracy.

In Bayesian statistics (see e.g. D. S. Sivia: “Data Analysis—A BayesianTutorial”, Oxford University Press, 2006, ISBN-13: 978-0198568322 or ETJaynes: “Probability Theory”, Cambridge University Press, 2003, ISBN-13:978-0521592710), information is always described in the form ofprobability distributions (in the case of continuous parameters it isprobability densities).

In this sense, a probability or probability density can be assigned toan individual eye model with a given set of parameters. Individual eyemodels that are consistent with the available information (e.g.objective wavefront measurement and biometrics of the eye) have a higherprobability or probability density, because e.g. the propagation andrefraction of a wavefront that a point light source would generate onthe retina after exiting the eye reproduces the measured data wellwithin the scope of the measurement accuracy of the objective wavefrontmeasurement, and likewise the parameters of the individual eye matchwith the available information about the biometry of the eye within thescope of the distributions known e.g. from the literature.

Individual eye models that are not consistent with the availableinformation are correspondingly assigned to low probabilities orprobability densities. The probability or probability density of anindividual eye model can be written as

prob(

_(i) |d _(i) ,I)

where

_(i) denotes the parameters of the individual eye model i, and d_(i) arethe measured data (it can e.g. include the current refraction or therefraction measured prior to eye surgery, the measured shape and/orrefractive properties of the cornea, the measured eye length or othervariables measured on the individual eye). With I, the current state ofknowledge upon evaluation of the data, i.e. the existing backgroundinformation (e.g. about the measurement process of the refraction, thedistribution of the parameters of the individual eye model or otherrelated variables in the population) is summarized. The vertical line,|′ means that the distribution of the variables to the left of ,|′ ismeant for given (i.e. fixed) variables to the right of ,|′.

The information obtained in the measurement process, in which the datad_(i) is measured, can also be understood as the probabilitydistribution of the data d_(i) with given parameters

_(i) of the individual eye model i:

prob(d _(i)|

_(i) ,I)

The accuracy of the measurement process is reflected in the width of thedistribution: an exact measurement has a narrower distribution than animprecise measurement, which has a wide or wider distribution of thedata d_(i).

Now, if one wishes to calculate the distribution of the parameters of anindividual eye with given data and background information, the followingproportionality can be used:

prob(

_(i) |d _(i) ,I)∝prob(

_(i) |I)prob(d _(i)|

_(i) ,I)

The term prob(

_(i)|I) describes the background knowledge about the parameters of theindividual eye model. This can be information from literature, forexample, but also information from data from past measurements. This canbe data from the same person for whom the ophthalmic lens is to bemanufactured, as well as data from measurements made for a large numberof other people.

The probability here serves as a measure of consistency. Parametervalues of the individual eye model that are consistent with themeasurements can be found in particular where both prob(

_(i)|I) and prob(d_(i)|

_(i), I) are high.

The probability or probability density prob(

_(i)|d_(i), I) can also be suitably normalized in order to write theproportionality as an equation.

The term prob(

_(i)|d_(i), I) can also include parameters of the eye lens. For example,some of the parameters

_(i) can include the refractive power of the eye lens, its positionand/or orientation in the eye, or other variables such as the refractiveindex and curvatures or shape of the surfaces.

The eye lens can be a natural lens. In this case, literature data on theparameters of natural eye lenses can be used (e.g. distributions of thecurvatures of the front and/or back surface, refractive index, etc.).

If the eye lens is an intraocular lens, the distributions of theparameters of natural eye lenses must not be used. Instead, theparameters of the intraocular lens should be used, provided they areindividually known. Otherwise, distributions of these parameters can beused from literature studies of eyes that underwent surgery. If suchinformation is not available, a flat distribution within reasonablelimits can be selected. For parameters that are positively definite anddefine length scales (e.g. radii of curvature or distances), it is alsopossible to select distributions that are flat in the logarithm of theseparameters.

Formally, the cases “natural eye lens” or “intraocular lens as eye lens”are to be described by different states of background knowledge I (i.e.I=I_(NL) bzw.I=I_(IOL)).

The probability or probability density prob prob(

_(i)|d_(i), I) can have one or more factors. Here, ach factor representsthe information about one or more parameters of the individual eyemodel. For example, the distribution of different independent parameters

_(i) ¹ and

_(i) ² from different literature sources can be represented as a product

prob(

_(i) |d _(i) ,I)=prob(

_(i) ¹,

_(i) ² |d _(i) ,I)=prob(

_(i) ¹ |d _(i) ,I)prob(

_(i) ² |d _(i) ,I).

By prob(

_(i)|I), parameters of the eye model can inadvertently be falsified. Forexample, if the “true” refraction is understood as a parameter of theindividual eye model, the most likely value of the “true” refraction candeviate from the measured refraction. If this is not desired, adistribution that is constant in the corresponding parameter (e.g.spherical equivalent of refraction) within carefully selected limits(e.g. between −30 dpt and +20 dpt for the spherical equivalent M, ±5 dptfor the astigmatic components J₀ and J₄₅) should be chosen.

If parameters of the individual eye model or other variables related tothe parameters or measurement data is known exactly or with a highdegree of accuracy, their distribution can be approximated as a Diracdelta distribution. The equations in these parameters or variables canbe integrated on both sides, which may simplify subsequent calculations.

Description of the Methods

Two possible methods for calculating an ophthalmic lens will bepresented below (Bayes A and Bayes B). In the Bayes A method, theavailable information is used to set up a (single) individual eye model,with the help of which an ophthalmic lens optimal for this eye model iscalculated. The eye model can e.g. be given by or assigned the set ofparameters

_(i) ^(max), which maximizes the probability or probability densityprob(

_(i)|d_(i),I). Other sets of parameters can also be selected, e.g. theexpected value <

_(i)> or the median

_(i) ^(med) of the parameters

_(i) with regard to the distribution prob(

_(i)|d_(i),I).

The Bayes B method is more advantageous—but computationally moredemanding—compared to Bayes A, since a subset of individual eye modelswith different sets of parameters can lead to ophthalmic lenses thathave very similar (even identical) properties (e.g. refractive power ina reference point of the ophthalmic lens, or the distribution of therefractive deficit across a given area of the ophthalmic lens, orsimilar criteria for determining the quality of an ophthalmic lens).Overall, an ophthalmic lens that was not calculated with the most likelyindividual eye model can therefore represent an optimal correction for asubset of individual eye models which overall have a higher probabilitythan the most likely individual eye model. It is therefore advantageousto search for the ophthalmic lens that optimally corrects thedistribution of eye models, instead of just determining the most likelyindividual eye model and manufacturing an ophthalmic lens for it.

In both methods, an ophthalmic lens (e.g. a spectacle lens) consistentwith the information available can be calculated.

Bayes A Method

In particular, one or more of the following steps can be carried out:

-   -   Providing an initial distribution of parameters of an eye model        (ideally as a multivariate probability distribution of all        parameters of the eye model, possibly also marginal        distributions; the probability distribution corresponds to the        information about the distribution of the parameters of the eye        model in the population of the persons);    -   Providing already known (in the best case measured) data on        properties of an individual eye (ideally with probability        distribution or measurement errors; the probability distribution        corresponds to the inaccuracy of the measurements) The already        known data can include: already known current subjective and/or        objective refraction, already known previous subjective and/or        objective refraction (e.g. prior to surgery), power and/or shape        and/or position (most important is the axial position) of        certain refractive surfaces of the eye, size and/or shape and/or        position of the entrance pupil, refractive index of the        refractive media, refractive index profile in the refractive        media, opacity; possibly determination of these variables        depending on the accommodation of the eye on a fixation object        (target) at a given close distance;    -   Determining the parameters of an individual eye model based on        the initial distribution of the parameters of the eye model and        the already known or measured data on the individual eye using        probability calculation. Ideally, the probability distribution        or, for example, the set of parameters characterizing a maximum        of the probability distribution is determined.    -   In particular, calculation methods such as Markov Chain Monte        Carlo, Variational Inference, Maximum Likelihood, Maximum        Posterior, or Particle Filter can be used;    -   The aim here is to select the parameters of the individual eye        model that are consistent both with the initial distribution of        eye models provided and with the data that is already known. The        product of the probability or probability density of the data        with given parameters of the eye model with the probability or        probability density of the parameters of the eye model is used        as the consistency measure.    -   Calculating/optimizing/selecting an ophthalmic lens in which at        least one parameter of the individual eye model is used.

The initial distribution of the parameters of eye models provided in thefirst step can be in a parameterized form, e.g. (possibly multivariate)normal distribution, other distribution of the exponential family,Cauchy distribution, Dirichlet process, etc., or as a set of samples,i.e. one or more (possibly multidimensional) data sets. If the initialdistribution of the parameters of eye models is parameterized, theparameters of this distribution are called “hyperparameters”.

The third step (i.e. determining the parameters of an individual eyemodel) can include determining a multivariate probability distributionthat includes both the parameters of the individual eye model and thehyperparameters of the initial distribution of the parameters of the eyemodel. In order to calculate the distribution of the parameters of theindividual eye model from this, the distribution must be marginalized,i.e. it is integrated via the hyperparameters. The integrals can besolved with common numerical methods (e.g. using Markov Chain MonteCarlo or Hybrid Monte Carlo) and/or analytical methods. The probabilityor the probability density of the parameters of the eye model can inthis case be calculated using the following equation:

prob(

_(i) |d _(i) ,I)∝prob(d _(i)|

_(i) ,I)∫dλprob(

_(i) ,λ|I)=prob(d _(i)|

_(i) ,I)∫dλprob(

_(i) |λ,I)prob(λ|I)

Here prob(

_(i)|λ,I) denotes the probability or probability density to find theparameters of the individual eye model

_(i) in the population characterized by the hyperparameters λ. Theintegrals are to be carried out over the entire definition ranges of allhyperparameters λ.

Bayes B Method

As an alternative or in addition to the Bayes A method, one or more ofthe following steps can be carried out:

-   -   Providing the distribution of at least one parameter of an        individual eye model;    -   Calculating the probability distribution of the parameters of        virtual ophthalmic lenses or calculating an ensemble of        ophthalmic lenses by optimizing/calculating/selecting virtual        ophthalmic lenses using at least one parameter of the individual        eye model;    -   Manufacturing an ophthalmic lens, with the aim that the        manufactured parameters of the ophthalmic lens achieve the        parameters of the virtual ophthalmic lens with the highest        probability.

In the first step, the distribution calculated analogously to steps 1 to3 of the Bayes A method can be provided. In the second step, the mostlikely parameters L_(i) of the ophthalmic lens are determined, i.e. onthe basis of the probability distribution or probability density

$\begin{matrix}{{{prob}\left( {\left. L_{i} \middle| d_{i} \right.,I} \right)} = {\int{d\vartheta_{i}{{prob}\left( {\vartheta_{i},\left. L_{i} \middle| d_{i} \right.,I} \right)}}}} \\{= {\int{d\vartheta_{i}{{prob}\left( {\left. L_{i} \middle| \vartheta_{i} \right.,I} \right)}{{prob}\left( {\left. \vartheta_{i} \middle| d_{i} \right.,I} \right)}}}} \\{= {\int{d\vartheta_{i}{\delta\left( {L_{i} - {L\left( \vartheta_{i} \right)}} \right)}{{prob}\left( {\left. \vartheta_{i} \middle| d_{i} \right.,I} \right)}}}}\end{matrix}$

the parameters of the ophthalmic lens L_(i) ^(max) are determined, whichmaximize prob(L_(i)|d_(i),I). Here, L_(i) initially denotes theparameters of any ophthalmic lens, and in the case of L_(i)=L(

_(i)) the parameters of the ophthalmic lens created when an ophthalmiclens is optimized with the aid of an individual eye model with theparameters

_(i). The Dirac delta distribution is denoted by δ(.).

The parameters of the ophthalmic lens can be e.g. vertex depth,refractive power at a reference point of the ophthalmic lens, refractivepower distribution over an area of the ophthalmic lens, refractiveerrors at a reference point of the ophthalmic lens, or the distributionof the refractive errors over an area of the ophthalmic lens.

It is important that the function L(

_(i)) can be non-linear, and therefore the maximum of the probabilitydensity prob(

_(i)|d_(i), I) (with respect to

_(i)) with L(

_(i)) is not necessarily mapped to the maximum of the probabilitydensity prob(L_(i)|d_(i),I).

If the function L(

_(i)) can be inverted piece by piece, the equation described above canalso be solved with the help of partial integration. Other methods arealso possible, e.g. numerical methods such as Particle Filter, MarkovChain Monte Carlo, or methods of parametric inference, with which adistribution of the parameters of the ophthalmic lens L_(i) can becalculated.

Both in the Bayes A method and in the Bayes B method, independent ofboth the number and type of variables known by measurement (i.e. thedata d_(i) and the form of the likelihood prob(d_(i)|

_(i), I)) as well as the number and type of parameters of the eye model

_(i), there always results a consistent eye model (Bayes A and B method)and possibly a choice of the parameters of the ophthalmic lens (Bayes Bmethod) that matches the ensemble of possible consistent eye models.

Examples Based on Probability Considerations to Solve Inconsistencies

Background to the Maximum Likelihood Approach

Basic Procedure

The initial situation is that N parameters x_(i), 1≤i≤N of a model areto undergo assignment and that the following information is available:

-   -   Mean values μ_(i), standard deviations σ_(i), und correlation        coefficients ρ_(ij) (with 1≤i, j≤N) of these N parameters in the        population;    -   Either there are no measurement values (k=0), or there are        measurement values x_(i) ^(mess), 1≤i≤k for k of these        parameters (where 1≤k≤N). The probability distribution for the        measurement value x_(i) ^(mess) of each parameter x_(i) is        described by a random variable X_(i). A reliability measure is        preferably available for each measurement value, e.g. a standard        deviation σ_(i) ^(mess) of the random variable X_(i), 1≤i≤k    -   For q=N−k of these parameters there are no measurement values.    -   Overall, only K of the N parameters are independent since the        model requires consistency conditions that can be expressed by        Q=N−K constraints.

Examples can be:

-   -   Example without HOA        -   Parameters (N=15): cornea (SZA), lens front surface (SZA),            lens back surface (SZA), vision disorder (SZA), eye length,            lens thickness, anterior chamber depth;        -   Measurement data (k=13): cornea (SZA), lens front surface            (SZA), lens back surface (SZA), vision disorder (SZA),            anterior chamber depth;        -   Constraints (Q=3): vision disorder (SZA)=theoretical vision            disorder (SZA) (calculated from the eye model assigned);    -   Example with HOA (up to radial order n=6)        -   Parameters (N=103): cornea (SZA+HOA), lens front surface            (SZA+HOA), lens back surface (SZA+HOA), vision disorder            (SZA+HOA), eye length, lens thickness, anterior chamber            depth;        -   Measurement data (k=101): cornea (SZA+HOA), lens front            surface (SZA+HOA), lens back surface (SZA+HOA), vision            disorder (SZA+HOA), anterior chamber depth;        -   Constraints (Q=25): vision disorder (SZA+HOA)=theoretical            vision disorder (SZA+HOA) (calculated from the eye model            assigned).

The basic problem to be solved is that in the case of measurement valuesthat deviate from the population mean, a decision must be made as towhether the measurement must be discarded (e.g. if it is implausible) ormust be adopted. If all measurement values are plausible in themselves,but violate one of the consistency conditions, then they must not all beadopted. Instead, a balance between the various measurement values mustthen be sought: those that have a very high measurement reliabilityshould at least almost be retained, while uncertain measurement valuesare more likely to be adapted. Preferably, the best possible values forall N parameters are identified from the known information.

The inventive idea is based in particular on the assumption that theparameters have certain (unknown but initially fixed) values. Under thisassumption, in the light of the above-mentioned information (statisticalvariables from the population, reliability measures of themeasurements), the conditional probability density

P ^(bed)(X ₁ , . . . ,X _(N) |x ₁ , . . . ,x _(N))  (1)

is established for the outcome of the measurements, where X₁, . . . ,X_(N) are the random variables that vary for fixed given true values x₁,. . . , x_(N). Subsequently, the probability for the observedmeasurement values P^(par) is then quantified by evaluating the functionP^(bed) for the k measurement values and marginalizing it for theremaining q=N−k (non-measured) parameters:

P ^(par)(x ₁ , . . . ,x _(N)):=∫P ^(bed)(X ₁ =x ₁ ^(mess) , . . . ,X_(k) =x _(k) ^(mess) ,X _(k+1) , . . . ,X _(N) |x ₁ , . . . ,x _(N))dX_(k+1) , . . . ,dX _(N)  (2)

This probability density is understood as a function P^(par)(x₁, . . . ,x_(N)) of the assumed N parameters x₁, . . . , x_(N). Those N parametervalues for which this function assumes a maximum are then preferablyconsidered to be the best possible values (maximum likelihood approach):

$\begin{matrix}{{\frac{\partial{P^{par}\left( {x_{1},\ldots,x_{N}} \right)}}{\partial x_{i}} = 0},{1 \leq i \leq N}} & (3)\end{matrix}$

As an alternative to the marginalization in Eq. (2), the N parametervalues can also be defined by setting the last parameters equal to themean values of the population,

x _(i)=μ_(i) ,k+1≤i≤N  (4)

while the first k parameters x₁, . . . , x_(k) are determined such thattheir expected values are equal to the measurement values:

X _(i) |x ₁ , . . . ,x _(k) ,x _(k+1)=μ_(k+1) , . . . ,x _(N)=μ_(N)

=x _(i) ^(mess)  (5)

As a further alternative, instead of the maximum formation according toequation (3) or the expected value formation according to equation (5),the medians can be used as a criterion as well.

As a further alternative, the maximum formation according to equation(3) and the expected value formation according to equation (5) as wellas the median determination can also be combined as desired in order todetermine the N parameter values.

Background to the Maximum Posterior Approach

The prior knowledge about the population is described by thedistribution P^(pop)(x₁, . . . , x_(N)), which can correspond to theprior of the Bayesian description. The total probability density, whichdescribes both the distribution of measurement values and of modelparameters, is thus given by the distribution function

P ^(ges)(X ₁ , . . . ,X _(k) ,x ₁ , . . . ,x _(N))=P ^(mess)(X ₁ , . . .,X _(k) |x ₁ , . . . ,x _(N))×P ^(pop)(x ₁ , . . . ,x _(N))  6

which can correspond to the posterior of the Bayesian description exceptfor one constant. This is why this approach is also referred to asmaximum posterior.

Preferably, P^(pop) is described by the multivariate normal distribution

$\begin{matrix}{{P^{pop}(x)} = {\sqrt{\frac{\det\left( C^{- 1} \right)}{\left( {2\pi} \right)^{N}}}{{Exp}\left( {{- \frac{1}{2}}\left( {x - µ} \right)^{T}{C^{- 1}\left( {x - µ} \right)}} \right)}}} & (7)\end{matrix}$

where μ is the vector of the mean values and C is the covariance matrix:

$\begin{matrix}{{µ = \begin{pmatrix}\mu_{1} \\\mu_{2} \\ \vdots \\\mu_{n}\end{pmatrix}},{C = \begin{bmatrix}\sigma_{1}^{2} & {\rho_{12}\sigma_{1}\sigma_{2}} & \ldots & {\rho_{1n}\sigma_{1}} \\{\rho_{12}\sigma_{1}\sigma_{2}} & \sigma_{2}^{2} & & {\rho_{2n}\sigma_{2}\sigma_{n}} \\ \vdots & & \ddots & \vdots \\{\rho_{1n}\sigma_{1}\sigma_{n}} & {\rho_{2n}\sigma_{2}\sigma_{n}} & \ldots & \sigma_{n}^{2}\end{bmatrix}}} & \left( {7a} \right)\end{matrix}$

The measurement is described by the distribution P^(mess)(X₁, . . . ,X_(k)|x₁, . . . ,x_(N)). Preferably, the measurements are independent

P ^(mess)(X ₁ , . . . ,X _(k) |x ₁ , . . . ,x _(N))=P ₁ ^(mess)(X ₁ |x₁) . . . P _(k) ^(mess)(X _(k) |x _(k))  (8)

The entire distribution function P^(ges) (except for the posteriorprefactor) is then given by

P ^(ges)(X ₁ , . . . ,X _(k) ;x ₁ , . . . ,x _(N))=P ₁ ^(mess)(X ₁ |x ₁). . . P _(k) ^(mess)(X _(k) |x _(k))×P ^(pop)(x ₁ , . . . x _(N))  (8a)

Particularly preferably, each of the measurements is normallydistributed with expected value x_(i) and standard deviation σ_(i)^(mess)

$\begin{matrix}{{P_{i}^{meas}\left( X_{i} \middle| x_{i} \right)} = {\frac{1}{\sqrt{2\pi}\sigma_{i}^{meas}}{{Exp}\left( {{- \frac{1}{2\left( \sigma_{i}^{meas} \right)^{2}}}\left( {X_{i} - x_{i}} \right)^{2}} \right)}}} & \left( {8b} \right)\end{matrix}$

The entire distribution function P^(ges) is then given by inserting thenormal distribution from Eq. (8b) into Eq. (8a).

It is the inventive idea to maximize P^(ges) as a function of theparameters x₁, . . . ,x_(N). In order to apply the maximum posteriorcriterion, it is preferred to form the derivatives of the logarithm

$\begin{matrix}{{{{\frac{\partial}{\partial x_{i}}\log}{P^{ges}\left( {X_{1},\ldots,X_{k},x_{1},{\ldots x_{N}}} \right)}} = {{\frac{\partial}{\partial x_{i}}\left\lbrack {{\log{P^{mess}\left( {X_{1},\ldots,\left. X_{k} \middle| x_{1} \right.,{\ldots x_{N}}} \right)}} + {\log{P^{pop}\left( {x_{1},\ldots,x_{N}} \right)}}} \right\rbrack} = {\frac{{\partial{P^{mess}\left( {X_{1},\ldots,\left. X_{k} \middle| x_{1} \right.,{\ldots x_{N}}} \right)}}/{\partial x_{i}}}{P^{mess}\left( {X_{1},\ldots,\left. X_{k} \middle| x_{1} \right.,{\ldots x_{N}}} \right)} + \frac{\partial{P^{pop}\left( {x_{1},\ldots,x_{N}} \right)}}{P^{pop}\left( {x_{1},\ldots,x_{N}} \right)}}}},{1 \leq i \leq N}} & (9)\end{matrix}$ $\begin{matrix}{{{\frac{\partial}{\partial x_{i}}\log}{P^{ges}\left( {X_{1},\ldots,X_{k},x_{1},{\ldots x_{N}}} \right)}} = {\frac{\partial}{\partial x_{i}}{{{\left\lbrack {{\log{P_{1}^{mess}\left( X_{1} \middle| x_{1} \right)}} + \ldots + {\log{P_{k}^{mess}\left( X_{k} \middle| x_{k} \right)}} + {\log{P^{pop}\left( {x_{1},\ldots,x_{N}} \right)}}} \right\rbrack = {\frac{{\partial{P_{1}^{mess}\left( X_{1} \middle| x_{i} \right)}}/{\partial x_{i}}}{P_{i}^{mess}\left( X_{1} \middle| x_{i} \right)} + \frac{\partial{P^{pop}\left( {x_{1},\ldots,x_{N}} \right)}}{P^{pop}\left( {x_{1},\ldots,x_{N}} \right)}}},{1 \leq i \leq N}}}}} & (10)\end{matrix}$

If the distributions are multivariate normally distributed, as isparticularly preferred, equation (9) or equation (10) represents alinear system of equations with N equations and N variables that can besolved for x₁, . . . , x_(N).

a) No Constraints

If there are no constraints and equation (9) can be solved, then uniquesolutions for x₁, . . . , x_(N) result. If the distributions aremultivariate normally distributed, as is particularly preferred, and ifthe measurement uncertainties are significantly smaller than the rangesof variation of the population, σ_(i) ^(mess)<<σ_(i), 1≤i≤k, then thesolutions result

x _(i) ≈x _(i) ^(mess),1≤i≤k

x _(i)≈μ_(i) +Δx _(i) ,k+1≤i≤N  (11)

i.e. for all parameters for which measurement values are available, oneessentially believes the measurement values, and for the remainingvalues one obtains the mean values μ_(i) of the population plus shiftsΔx_(i) due to the correlations with the measurement values. Oneembodiment of the invention then consists in adopting the measurementvalues for 1≤i≤k directly and neglecting their slight shift due to theunderlying population.

b) Constraints

If there are constraints between the parameters, then every member ofthe population also satisfies these constraints. Constraints can bedescribed by

f _(j)(x ₁ , . . . ,x _(N))=0,1≤j≤Q⇔f(x ₁ . . . ,x _(N))=0  (12)

i.e. by Q functions f_(j) of the parameters x₁, . . . , x_(N), which canbe combined in a vector f and which, by requirement, are to be equal tozero. The functions f_(j) are preferably linear or linear approximationsto the given constraints.

In the preferred case of multivariate distributions, this has theconsequence that the columns of the covariance matrix are linearlydependent, that is to say that the covariance matrix has a rank r<N andcan therefore no longer be inverted. A distribution density P^(pop)(x₁,. . . , x_(N)) can then no longer be specified.

One possibility in practice is to regularize the covariance matrix C byshifting one or more of the correlations ρ_(ij) or standard deviationsσ_(i) contained in it by ε and then determining x₁, . . . , x_(N). Thesolutions thus obtained then automatically satisfy the constraints forε→0.

In the context of the invention, it has been found that this method hasdisadvantages though. On the one hand, one has to know the distributionin the population, and on the other hand, its covariance matrix iseither singular or poorly conditioned. If one researches thecorrelations ρ_(ij) and standard deviations σ_(i), then smallinaccuracies in the information or incomplete information are sufficientfor the covariance matrix to be regular, but then possibly generatenumerically unstable solutions for the parameters sought.

In the context of the invention, however, it has been found that thisproblem can be circumvented by either working on the basis of thedistribution (maximum likelihood approach)

P ^(mess)(x ₁ , . . . ,x _(N)):=P ^(mess)(X ₁ =x ₁ ^(mess) , . . . ,X_(k) =x _(k) ^(mess) |x ₁ , . . . ,x _(N))  (13a)

or on the basis of the distribution (maximum posterior approach)

P ^(ges)(x ₁ , . . . ,x _(N)):=P ^(ges)(X ₁ =x ₁ ^(mess) , . . . ,X _(k)=x _(k) ^(mess) ;x ₁ , . . . ,x _(N))  (13b)

The substitution method can preferably be used for this purpose.

Maximum Likelihood Method with Constraints and Substitution

The first K parameters x″:=(x₁, . . . , x_(K))^(T) are assumed to beindependent and equation (12) is solved for the remaining Q=N−Kdependent parameters x^(a):=(x_(K+1), . . . , x_(N))^(T), which can thenbe understood as a function x^(a)(x^(M)) of the independent parametersx^(M) and can be substituted in f. Then the constraints as a function ofx^(u) are:

f(x ^(u) ,x ^(a)(x ^(u)))=0  (14).

In the context of the invention, it is not necessary to explicitly knowthe function x^(a) (x^(u)). In the context of the invention, one onlyneeds its Jacobi matrix ∂x^(a)/∂x^(u):=∂x_(i) ^(a)/∂x_(j) ^(u), 1≤i≤Q,1≤j≤K, which according to the theorem of the implicit function is givenby

$\begin{matrix}{\frac{\partial x^{a}}{\partial x^{\mu}} = {{- \left( \frac{\partial f}{\partial x^{a}} \right)^{- 1}}\frac{\partial f}{\partial x^{u}}}} & (15)\end{matrix}$

where ∂f/∂x^(a) is the quadratic Jacobi matrix of f with regard to x^(a)and ∂f/∂x^(u) is the generally rectangular Jacobi matrix of f withregard to x^(u). The probability density P^(mess)(x^(u), x^(a)(x^(u)))thus has to be maximized as a function of x^(u), i.e.

$\begin{matrix}{\begin{matrix}{{\frac{\partial}{\partial x^{u}}{P^{mess}\left( {x^{u},{x^{a}\left( x^{u} \right)}} \right)}} = {\frac{\partial P^{mess}}{\partial x^{u}} + {\frac{\partial P^{mess}}{\partial x^{a}}\frac{\partial x^{a}}{\partial x^{u}}}}} \\{= {\frac{\partial P^{mess}}{\partial x^{u}} - {\frac{\partial P^{mess}}{\partial x^{a}}\left( \frac{\partial f}{\partial x^{u}} \right)^{- 1}\frac{\partial f}{\partial x^{u}}}}} \\{= 0}\end{matrix}.} & (16)\end{matrix}$

The system of equations (16) is K equations that can be solved for theparameters x^(u) independent for K. The remaining parameters x^(a) areobtained by inserting them into the context x^(a)(x^(u)).

Maximum Likelihood Method with Constraints and Lagrange Parameters

Alternatively, within the scope of the invention, the entire set ofparameters can be considered independent if, instead of the functionP^(mess)(x₁, . . . , x_(N)), the Lagrange function is maximized

P ^(mess,Lagrange)(x ₁ , . . . x _(N),λ)=P ^(mess)(x ₁ , . . . ,x_(N))+λf(x ₁ , . . . ,x _(N))  (17)

where λ=(λ₁, . . . , λ_(Q)) is a Q-dimensional vector of Lagrangemultipliers. It is then to be maximized by setting the derivatives N+Qto zero

$\begin{matrix}{{{\frac{\partial}{\partial x_{i}}{P^{{mess},{Lagrange}}\left( {x_{1},\ldots,{x_{N}\lambda}} \right)}} = 0},{1 \leq i \leq N}} & (18)\end{matrix}$${{\frac{\partial}{\partial\lambda_{j}}{P^{{mess},{Lagrange}}\left( {x_{1},\ldots,{x_{N}\lambda}} \right)}} = 0},{1 \leq j \leq Q}$

Solving equation (18) for the N+Q unknowns (x₁, . . . , x_(N)) and (λ₁,. . . , λ_(Q)) leads to the solutions for the parameters.

Instead of treating the constraints with substitution or long-rangeparameters, one can alternatively (for example in the case of locallyvanishing gradients of the function to be maximized) use a dampedHamilton formalism with a friction term.

Analogously, the method of Eqs. (16) to (18) can be applied to thefunction P^(ges)(x₁, . . . , x_(N)) instead of P^(mess)(x₁, . . . ,x_(N)) and then represents a maximum-posterior method with constraints.

Embodiment with Specific Exemplary Numerical Values

For the sake of simplicity, an eye that is rotationally symmetricalabout the optical axis and therefore has neither a cylindricalprescription, nor a cylindrical cornea, nor cylindrical lens surfaces isconsidered as a starting situation. Exemplary values and parametersprior to the IOL surgery are in detail:

S=−7.0 dpt;vision disorder (measured)

C=41.2 dpt;refractive power of cornea (measured)

L ₁=7.82 dpt;refractive power of lens front surface (literature)

L ₂=13.28 dpt;refractive power of lens back surface (literature)

d _(CL)=3.6 mm;anterior chamber depth (measured)

d _(L)=3.7 mm;lens thickness (literature)

n _(CL)=1.336;refractive index anterior chamber (literature)

n _(L)=1.422;refractive index lens (literature)

n _(LR)=1.336;refractive index vitreous body (literature)  (19).

After IOL surgery, for example the following values or parameters aretransmitted:

S _(IOL) ^(mess)=0.0 dpt;vision disorder(measured)

L _(2,IOL) ^(mess)=3.2 dpt;refractive power of lens back surface(manufacturer information)  (20).

All other parameters after IOL surgery are assumed to be unchanged forthe sake of simplicity.

With the aid or equation

$\begin{matrix}{D_{LR} = {L_{2} + {\left( {L_{1} + \frac{S + C}{1 - {\tau_{CL}\left( {S + C} \right)}}} \right)\left( {1 - {\tau_{L}\left( {L_{1} + \frac{S + C}{1 - {\tau_{CL}\left( {S + C} \right)}}} \right)}} \right)^{- 1}}}} & (21)\end{matrix}$

one can calculate the reduced inverse vitreous length(D_(LR)=n_(LR)/d_(LR), where d_(LR) the vitreous length is; furtherτ_(CL)=d_(CL)/n_(CL) and τ_(L)=d_(CLL)/n_(L)), and thus the eye lengthd_(A)=d_(CL)+d_(L)+d_(LR). Vitreous body length and eye length are sodirectly related that in the following the vitreous body length can beconsidered instead of the eye length.

If one applies equation (21) to the situations before and after surgery,one formally obtains prior to IOL surgery

D _(LR)=64.69 dpt  (22a)

and formally after IOL surgery

D _(LR,IOL) ^(mess)=65.65 dpt  (22b).

However, since the vitreous body length cannot have changed as a resultof surgery, there is an inconsistency here that can be solved within thescope of the present invention.

In order to choose the simplest possible example, the initial situationcan be regarded as the case that there are no variations and nocorrelations in the basic population, and that only the vision disordermeasured afterward as well as the IOL itself are subject touncertainties:

σ_(S,IOL) ^(mess)=0.25 dpt;vision disorder (Std .deviation ofmeasurement method)

σ_(2,IOL) ^(mess)=0.4 dpt;refractive power of lens back surface(manufacturer tolerance)  (23).

Now, within the scope of the invention, the true values of S_(IOL),L_(2,IOL) can be identified which, as expected, will both deviate fromequation (20).

In the exemplary case, P^(pop)≡1 and the probability density for thedistribution of S_(IOL), L_(2,IOL) to be initially assumed based on themeasurements is

$\begin{matrix}{{P^{mess}\left( {S_{IOL}^{mess},\left. L_{2,{IOL}}^{mess} \middle| S_{IOL} \right.,L_{2,{IOL}}} \right)} = {{{P_{1}^{mess}\left( S_{IOL}^{mess} \middle| S_{IOL} \right)}\ldots{P_{k}^{mess}\left( L_{2,{IOL}}^{mess} \middle| L_{2,{IOL}} \right)}} = {\frac{1}{2\pi\sigma_{S,{IOL}}^{mess}\sigma_{{L2},{IOL}}^{mess}} \times {{Exp}\left( {{- \frac{1}{2\left( \sigma_{S,{IOL}}^{mess} \right)^{2}}}\left( {S_{IOL} - S_{IOL}^{mess}} \right)^{2}} \right)} \times {{{Exp}\left( {{- \frac{1}{2\left( \sigma_{{L2},{IOL}}^{mess} \right)^{2}}}\left( {L_{2,{IOL}} - L_{2,{IOL}}^{mess}} \right)^{2}} \right)}.}}}} & (24)\end{matrix}$

Now, however, the constraint applies that S_(IOL), L_(2,IOL) afterinsertion in equation (21) have to yield the same value for D_(LR) aftersurgery as before surgery. Hence the equation for the constraint is

$\begin{matrix}{D_{LR} = {L_{2,{IOL}} + {\left( {L_{1} + \frac{S_{IOL} + C}{1 - {\tau_{CL}\left( {S_{IOL} + C} \right)}}} \right)\left( {1 - {\tau_{L}\left( {L_{1} + \frac{S_{IOL} + C}{1 - {\tau_{CL}\left( {S_{IOL} + C} \right)}}} \right)}} \right)^{- 1}}}} & (25)\end{matrix}$

which when solved for L_(2,IOL) yields as a function of S_(IOL):

$\begin{matrix}{{L_{2,{IOL}}\left( S_{IOL} \right)} = {D_{LR} - {\left( {L_{1} + \frac{S_{IOL} + C}{1 - {\tau_{CL}\left( {S_{IOL} + C} \right)}}} \right){\left( {1 - {\tau_{L}\left( {L_{1} + \frac{S_{IOL} + C}{1 - {\tau_{CL}\left( {S_{IOL} + C} \right)}}} \right)}} \right)^{- 1}.}}}} & (26)\end{matrix}$

The constraint means that one may only move on the cutting surface 30shown in FIG. 2 .

If one substitutes L_(2,IOL)(S_(IOL)) in equation (24) and maximizes forS_(IOL), i.e. if one solves

$\begin{matrix}{{\frac{d}{{dS}_{IOL}}{P^{mess}\left( {S_{IOL}^{mess},\left. L_{2,{IOL}}^{mess} \middle| S_{IOL} \right.,{L_{2,{IOL}}\left( S_{IOL} \right)}} \right)}} = 0} & (27)\end{matrix}$

for S_(IOL), one obtains

S _(IOL)=−0.12 dpt

L _(2,IOL)(S _(IOL))=3.01 dpt  (28).

Both variables S_(IOL), L_(2,IOL) are therefore in the negativedirection compared to the measurement values, but not to the sameextent. Rather, the method seeks a balance in the light of the differentstandard deviations and the asymmetrical position of the constraintrelative to the Gaussian bell.

Inconsistencies in the eye model can occur not only for a calculated eyelength (or a calculated lens-retina distance), but also e.g. whenmeasuring the eye length. Such inconsistencies can be solved analogouslyto the example of a calculated eye length described above. Of course,more complex examples in which the eye length itself is also not fixedor where possibly correlations occur, can also be given.

First example of a Bayes A method The eye model that is used in thisexample consists of a vision disorder, c_(n) ^(m), described up to the4^(th) Zernike order, which relates to a pupil diameter of 5 mm, as wellas the natural logarithm of a pupil radius log r_(ph) or log r_(mes)present under photopic or mesopic lighting conditions. Overall, themodel parameters of the eye model can be written as a vector

_(i)=(log r _(ph),log r _(mes) ,c ₂ ⁻² ,c ₂ ⁰ ,c ₂ ⁺² ,c ₃ ⁻³ ,c ₃ ⁻¹ ,c₃ ⁺¹ ,c ₃ ⁺³ ,c ₄ ⁻⁴ ,c ₄ ⁻² ,c ₄ ⁰ ,c ₄ ⁺² ,c ₄ ⁺⁴)

The Zernike coefficients of the 0^(th) to 1^(st) order (piston, and thehorizontal and vertical prism) were not considered here, as they do notplay a role in the vision disorder of the eye and can be assumed to beconstantly 0, for example.

The measurement data d_(i) known for an individual eye are here sphere,cylinder and axis of the (far) refraction Rx,

Rx=(S ^(Rx) ,Z ^(Rx) ,A ^(Rx))=(2.5 dpt,−1 dpt,100°),

which written as a power vector is

P ^(Rx)=(M ^(Rx) ,J ₀ ^(Rx) ,J ₄₅ ^(Rx))=(2.00 dpt,−0.47 dpt,−0.17 dpt)

The measurement error of the refraction is also known and as thestandard deviation in the individual power vector components is

σ_(P) ^(Rx)=(σ_(M) ^(Rx),σ_(J0) ^(Rx),σ_(J45) ^(Rx))=(0.375 dpt,0.125dpt,0.125 dpt).

It is assumed that the measurement error is normally distributed as apower vector around the power vector of the vision disorder P^(eye)(

_(i)) that can be identified from the model parameters, so that

prob(P ^(Rx)|

_(i) ,I)=prob(P ^(Rx) |P ^(eye)(

_(i)),I)=prob(M ^(Rx) ,J ₀ ^(Rx) ,J ₄₅ ^(Rx) |P ^(eye)(

_(i)),I)

can be written for the likelihood. Here, P^(eye)(

_(i)) is the power vector (M^(eye),J₀ ^(Rx),J₄₅ ^(eye)), which with thehelp of the Root-Mean-Squared (RMS) metric results from a Zernikewavefront scaled to the photopic pupil radius r_(ph) according to knownmethods. In this case, P^(eye) depends only on a part of the parametersof the eye model:

P ^(eye)(

_(i))=P ^(eye)(log r _(ph) ,c ₂ ⁻² ,C ₂ ⁰ ,C ₂ ⁺² ,C ₄ ⁻² ,C ₄ ⁰ ,C ₄⁺²).

The likelihood can now be written out as

${{prob}\left( {M^{Rx},J_{0}^{Rx},\left. J_{45}^{Rx} \middle| {P^{eye}\left( \vartheta_{i} \right)} \right.,\sigma_{P}^{Rx},I} \right)} = {\frac{1}{\left( {2\pi} \right)^{\frac{3}{2}}\sigma_{M}^{Rx}\sigma_{J0}^{Rx}\sigma_{J45}^{Rx}}{{\exp\left\lbrack {{- \frac{\left( {M^{Rx} - M^{eye}} \right)^{2}}{2\left( \sigma_{M}^{Rx} \right)^{2}}} - \frac{\left( {J_{0}^{Rx} - J_{0}^{eye}} \right)^{2}}{2\left( \sigma_{J0}^{Rx} \right)^{2}} - \frac{\left( {J_{45}^{Rx} - J_{45}^{eye}} \right)^{2}}{2\left( \sigma_{J45}^{Rx} \right)^{2}}} \right\rbrack}.}}$

In this example, the most likely eye model is identified for givenbackground knowledge I (distribution of the model parameters

_(i) in the population, measurement accuracy of the refraction,determination of the power vector from a vision disorder in Zernikerepresentation, etc.) and the known measurement data P^(Rx).

As a prior prob(

_(i)|I), approximate use was made of a sample of the model parametersthat was identified with the help of a measuring device (here anaberrometer) for a large number of people. The actual prior has 14partly dependent parameters and as such cannot be illustratedholistically. In FIGS. 3 a to 3 f , however, marginal densities of thesample from the prior used for the calculation are shown.

The posterior consists of the likelihood-weighted sample of the prior.To this end, the likelihood for each element (sample) of the sample ofthe prior was evaluated and used as a weight. Alternatively, a smallersample size can be used, in which case an unweighted sample must betaken from the weighted sample. Marginal posterior densities are to beseen analogously to the prior in FIGS. 4 a to 4 f.

The maximum of the posterior density,

_(i) ^(max), was approximated by means of a kernel density estimation(kernel: multivariate normal distribution with a standard deviation thatcorresponds to 0.5 times the standard deviation of the posteriordistribution of the parameters of the eye model). This resulted in thefollowing values for the most likely eye model (Zernike coefficients arerelated to a pupil of 5 mm diameter):

log r_(ph) ^(max)/mm log r_(mes) ^(max)/mm   0.406572126   0.819130806c₂ ^(−2, max)/μm c₂ ^(0, max)/μm c₂ ^(+2, max)/μm   0.241991418−1.544175270   0.507974951 c₃ ^(−3, max)/μm c₃ ^(−1, max)/μm c₃^(+1, max)/μm c₃ ^(+3, max)/μm −0.155674297 −0.014110217 −0.046864893  0.055671727 c₄ ^(−4, max)/μm c₄ ^(−2, max)/μm c₄ ^(0, max)/μm c₄^(2, max)/μm c₄ ^(4, max)/μm −0.015747182   0.003006757   0.074465687−0.005984227 0.035345112

The power of the lens, P_(mes)(ϑ_(i) ^(max)), which optimally correctsthe most likely eye model under mesopic lighting conditions, wascalculated by scaling the Zernike coefficients c₂ ^(−2,max), c₂^(0,max), c₂ ^(+2,max), c₃ ^(−3,max), c₃ ^(−1,max), c₃ ^(+1,max), c₃^(+3,max), c₄ ^(−4,max), c₄ ^(−2,max), c₄ ^(0,max), c₄ ^(+2,max), c₄^(+4,max), related to the pupil diameter of 5 mm to the most likelymesopic pupil with the radius r_(mes) ^(max) with the aid of the methodknown from the literature. From this vision disorder of the eye, stillgiven in the Zernike representation, the power vector was againidentified with the RMS metric

$\begin{matrix}{{P_{mes}\left( \vartheta_{i}^{\max} \right)} = \left( {{M_{mes}\left( \vartheta_{i}^{\max} \right)},{J_{0,{mes}}\left( \vartheta_{i}^{\max} \right)},{J_{45,{mes}}\left( \vartheta_{i}^{\max} \right)}} \right)} \\{= \left( {{1.77{dpt}},{{- 0.4}{dpt}},{{- 0.19}{dpt}}} \right)}\end{matrix}$

This can be used to produce a lens that optimally corrects the mostlikely eye model under mesopic lighting conditions (e.g. a single visionlens or as a power in the distance reference point of a progressivelens).

Analogously, the power of a lens that optimally corrects the most likelyeye model under photopic lighting conditions can be calculated byscaling the vision disorder in Zernike representation to the most likelyphotopic pupil with radius r_(ph) ^(max). This yields

$\begin{matrix}{{P_{ph}\left( \vartheta_{i}^{\max} \right)} = \left( {{M_{ph}\left( \vartheta_{i}^{\max} \right)},{J_{0,{ph}}\left( \vartheta_{i}^{\max} \right)},{J_{45,{ph}}\left( \vartheta_{i}^{\max} \right)}} \right)} \\{= {\left( {{1.92{dpt}},{{- 0.41}{dpt}},{{- 0.18}{dpt}}} \right).}}\end{matrix}$

In contrast to other known methods (see e.g. WO°2013°087212°A1), in themethod presented here, the finite measurement error of the data (herethe refraction) can be taken into account. This has the effect that thelikelihood as a function of the data has a finite width. For thisreason, the most likely power vector in case of a photopic pupil,P_(ph)(

_(i) ^(max)), is also shifted more toward the maximum of the priorcompared to the refraction (also carried out under photopic lightingconditions). This is an advantage compared to the method disclosed inWO°2013°087212°A1, since in this way the optimal correction for the eyeis made statistically more frequently. It is also not wrongly assumed(such as in regression analyzes) that parameters that are actually notexactly known have an exact value (this is true for the dependentvariables in regression analyzes), but the actual information (e.g. theactual or the estimated measurement accuracy) of the parameters involvedis taken into account.

First Example of a Bayes B Method

The eye model considered in this example and the measurement data (herethe refraction) are chosen as in the Bayes A example. Now, however, theaim is to calculate the most likely power of an ophthalmic lens to bemanufactured, and not the power of the ophthalmic lens for the mostlikely eye model. As in the example of the Bayes A method, the lens isto be ideally suited for mesopic vision. In this example, the parametersof the ophthalmic lens, L_(i), correspond to their power in simplifiedform: L_(i)=P^(L)=(M^(L), J₀ ^(L), J₄₅ ^(L)). For this purpose, mesopicZernike wavefronts were calculated from the posterior sample of ExampleBayes A by scaling to the respective mesopic pupil for the Zernikecoefficients related to a pupil diameter of 5 mm in this sample usingthe method known from the literature. Power vectors for the mesopicpupil were identified from these mesopic Zernike wavefronts using theroot-mean-squared (RMS) metric. These in turn represent a sample fromthe posterior distribution of the power of an ophthalmic lens optimizedfor a mesopic pupil (cf. FIGS. 5 a to 5 c ). The maximum of theposterior distribution was approximated by the maximum of a kerneldensity estimation of the sample (multivariate normal distribution askernel with a standard deviation corresponding to 0.5 times the standarddeviation of the posterior distribution of the power vector). Thisyielded the following most likely power

P _(mes) ^(L, max)=(M _(mes) ^(L, max) ,J _(0,mes) ^(L, max) ,J_(45,mes) ^(L, max))=(1.80 dpt,−0.41 dpt,−0.14 dpt)

of an ophthalmic lens to be manufactured (e.g. the power of a spectaclelens such as a single vision lens, or the power in the distancereference point of a progressive lens), which makes optimal use of theinformation available about the eye model. This most likely powerdiffers from the power P_(mes)(

_(i) ^(max)) calculated in the Bayes A example for the most likely eyemodel due to the non-linear transformation (here scaling) of theparameters of the eye model (here the vision disorder described up tothe 4^(th) Zernike order and the logarithms the mesopic and photopicpupils) into the parameters of the ophthalmic lens (here the power ofthe ophthalmic lens as a power vector). As a correction, P_(mes)^(L, max) in turn represents a further improvement compared to thecorrection P_(mes)(

_(i) ^(max)), since P_(mes) ^(L, max) corresponds to the most likelyvision disorder for the given information, and P_(mes)(

_(i) ^(max)) only to the vision disorder of the most likely eye modelbut generally not the most likely vision disorder.

For the person skilled in the art, other examples are easily feasible,in which the Bayes A or Bayes B methods are used, but in which a muchmore complex eye model is used, which e.g. can consist of severalrefractive surfaces and media with different refractive indices. Each ofthe surfaces can be described e.g. in a Zernike representation, thedistribution of the coefficients can be partially or fully described inthe literature or accessible through measurements (e.g. with the help ofmeasuring methods for determining eye biometry such as scanning opticalcoherence tomography, ultrasound or magnetic resonance tomography). Ifsuch information is missing, priors can be replaced with the help ofassumptions about the smoothness of the refracting surfaces or the localcurvature properties of these surfaces (e.g. correlation lengths of thelocal curvatures). The refractive indices of the media can also beincluded in the model as parameters that are not precisely known. Modelswith refractive index gradients are also possible. The propagation andrefraction of the light through the model eye is selected according tothe eye model used. Other known metrics (monochromatic or polychromatic)can also be used as metrics.

Other likelihood distributions can also be selected if this is motivated(e.g. other parameters of the eye model can be measured directly, orother measured variables that are indirectly related to the parametersof the eye model can be measured or otherwise determined, e.g. eyelength and/or distances between the refracting surfaces of the eye).

It is also easily possible to use more complex descriptions of the lens(e.g. through the course of the front and back surfaces and thethickness). If the complexity (i.e. the number of parameters) of the eyeand/or lens models is very high, the posterior can be solved e.g. byapproximation methods such as parametric inference (e.g. variationalinference), in which the posterior itself is parameterized and itsdetermination is understood as an optimization problem.

Second Example of a Bayes a Method

The eye model used in this example consists of three refractive surfacesdescribed in the second order and not tilted against each other (cornealfront surface, as well as front and back surface of the eye lens, herenumbered with surface k=1, 2 and 3), as well as the retina. The cornealback surface is neglected in this model (see Bennett-Rabbett's eyemodel). To parameterize the surface power of each surface, the surfacepower is used in the cross-section 0°, 45° and 90° to the horizontal(i.e. the elements of the surface power matrix, designations for thisare D_(xx) ^(k), D_(xy) ^(k) und D_(yy) ^(k)). The mutual position ofthe refractive surfaces and the retina is parameterized as a positivequantity by the natural logarithms of the distances between two directlyadjacent surfaces (logarithms of the distances cornea-lens frontsurface, lens front surface-lens back surface, and the lens backsurface-retina are designated with log d₁₂, log d₂₃ and log d₃₄,respectively, wherein the distances are used in mm). The refractiveindices of the media between the refractive surfaces (i.e. in theanterior chamber, eye lens, and vitreous body) have the refractiveindices n₁₂, n₂₃ und n₃₄, respectively. Here, the parameter vector issummarized as

_(i)=(log d ₁₂,log d ₂₃,log d ₃₄ ,D _(xx) ¹ ,D _(xy) ¹ ,D _(yy) ¹ ,D_(xx) ² ,D _(xy) ² ,D _(yy) ² ,D _(xx) ³ ,D _(xy) ³ ,D _(yy) ³ ,n ₁₂ ,n₂₃ ,n ₃₄)

The measurement data d_(i) known for an individual eye are here sphere,cylinder and axis of the (far) refraction Rx,

Rx=(S ^(Rx) ,Z ^(Rx) ,A ^(Rx))=(2.5 dpt,−1 dpt,100°),

which written as a power vector is

P ^(Rx)=(M ^(Rx) ,J ₀ ^(Rx) ,J ₄₅ ^(Rx))=(2.00 dpt,−0.47 dpt,−0.17 dpt)

The measurement error of the refraction is also known and as thestandard deviation in the individual power vector components is

σ_(P) ^(Rx)=(σM ^(Rx),σ_(J0) ^(Rx),σ_(J45) ^(Rx))=(0.375 dpt,0.125dpt,0.125 dpt).

It is assumed that the measurement error is normally distributed as apower vector around the power vector of the vision disorder P^(eye)(

_(i)) that can be identified from the model parameters, so that

prob(P ^(Rx)|

_(i) ,I)=prob(P ^(Rx) |P ^(eye)(

_(i)),I)=prob(M ^(Rx) ,J ₀ ^(Rx) ,J ₄₅ ^(Rx) |P ^(eye)(

_(i)),I)

can be written for the likelihood. Here, P^(eye)(ϑ_(i)) is the powervector (M^(eye))(

_(i)), J₀ ^(eye)(

_(i)), J₄₅ ^(eye)(

_(i))), which corresponds to the vision disorder of the eye model andcan be calculated by repeated propagation and refraction of a wavefront,emanating from a point on the retina, through the eye to the vertex ofthe cornea from the parameters of the eye model in a paraxialapproximation.

The likelihood can now be written out as

${{prob}\left( {M^{Rx},J_{0}^{Rx},\left. J_{45}^{Rx} \middle| {P^{eye}\left( \vartheta_{i} \right)} \right.,\sigma_{P}^{Rx},I} \right)} = {\frac{1}{\left( {2\pi} \right)^{\frac{3}{2}}\sigma_{M}^{Rx}\sigma_{J0}^{Rx}\sigma_{J45}^{Rx}}{{\exp\left\lbrack {{- \frac{\left( {M^{Rx} - {M^{eye}\left( \vartheta_{i} \right)}} \right)^{2}}{2\left( \sigma_{M}^{Rx} \right)^{2}}} - \frac{\left( {J_{0}^{Rx} - {J_{0}^{eye}\left( \vartheta_{i} \right)}} \right)^{2}}{2\left( \sigma_{J0}^{Rx} \right)^{2}} - \frac{\left( {J_{45}^{Rx} - {J_{45}^{eye}\left( \vartheta_{i} \right)}} \right)^{2}}{2\left( \sigma_{J45}^{Rx} \right)^{2}}} \right\rbrack}.}}$

In this example, the most likely eye model is identified for givenbackground knowledge I (distribution of the model parameters i in thepopulation, measurement accuracy of the refraction, determination of thepower vector from a vision disorder in Zernike representation, etc.) andthe known measurement data P^(Rx).

For the prior prob(

_(i)|I), a multivariate normal distribution was assumed, the parametersof which are derived from a series of measurements of the biometrics ofthe eyes of a large number of people, literature values, and estimatesof the scattering of the respective variables in the population (if noinformation on the scattering range range was found).

The following table shows the maximum and the standard deviation (inbrackets) of the normal distribution:

log d₁₂/mm log d₂₃/mm log d₃₄/mm  1.26 (0.0839)  1.33 (0.0856)  2.81(0.0688) D_(xx) ¹/dpt D_(xy) ¹/dpt D_(yy) ¹/dpt 44.25 (1.61)  0.00(0.21) 43.49 (1.47)  D_(xx) ²/dpt D_(xy) ²/dpt D_(yy) ²/dpt 7.82 (0.56)0.00 (0.25) 7.82 (0.56) D_(xx) ³/dpt D_(xy) ³/dpt D_(yy) ³/dpt 13.28(0.56)  0.00 (0.25) 13.28 (0.56)  n₁₂ n₁₂ n₃₄ 1.336 (0.001) 1.422(0.001) 1.336 (0.001)

The correlation matrix of the normal distribution had a diagonaloccupied by 1 and was occupied by zero everywhere except for thefollowing non-diagonal elements:

log d₁₂/ log d₂₃/ log d₃₄/ D_(xx) ¹/ D_(xy) ¹/ D_(yy) ¹/ mm mm mm dptdpt dpt log d₁₂/mm 1.000 −0.581 0.371 0.232 0.045 0.151 log d₁₂/mm−0.581 1.000 −0.254 −0.239 −0.132 −0.116 log d₁₂/mm 0.371 −0.254 1.000−0.250 −0.063 −0.443 D_(xx) ¹/dpt 0.232 −0.239 −0.250 1.000 0.131 0.863D_(xx) ¹/dpt 0.045 −0.132 −0.063 0.131 1.000 0.132 D_(xx) ¹/dpt 0.151−0.116 −0.443 0.863 0.132 1.000

The actual prior has 15 partly dependent parameters and as such cannotbe illustrated holistically. In FIGS. 6 a to 6 e , however, marginaldensities of the sample from the prior used for the calculation areshown as scatter diagrams.

The posterior consists of the likelihood-weighted sample of the prior.To this end, the likelihood for each element (sample) of the sample ofthe prior was evaluated and used as a weight. Alternatively, a smallersample size can be used, in which case an unweighted sample must betaken from the weighted sample. Marginal posterior densities are to beseen analogously to the prior in FIGS. 7 a to 7 e.

The maximum of the posterior density,

_(i) ^(max), was approximated by means of a kernel density estimation(kernel: multivariate normal distribution with a standard deviation thatcorresponds to 0.5 times the standard deviation of the posteriordistribution of the parameters of the eye model). This resulted in thefollowing values

_(i) ^(max) for the most likely eye model

log d₁₂ ^(max)/mm log d₂₃ ^(max)/mm log d₃₄/mm 1.202 1.372 2.768 D_(xx)^(1, max)/dpt D_(xy) ^(1, max)/dpt D_(yy) ^(1, max)/dpt 43.893  −0.081  42.914  D_(xx) ^(2, max)/dpt D_(xy) ^(2, max)/dpt D_(yy) ^(2, max)/dpt7.761 0.046 7.636 D_(xx) ^(3, max)/dpt D_(xy) ^(3, max)/dpt D_(yy)^(3, max)/dpt 13.223  0.087 13.226  n₁₂ ^(max) n₁₂ ^(max) n₃₄ ^(max)1.336 1.422 1.336

One can easily see from the differences in the xx and yy components ofthe surface powers that most of the astigmatism is present in thecornea, since this has the greatest fluctuation range in the powers inthe population (and therefore also in the prior).

The power of the lens that optimally corrects the most likely eye model,P(

_(i) ^(max)) was also calculated and is

P(

_(i) ^(max))=(M(

_(i) ^(max)),J ₀(

_(i) ^(max))J ₄₅(

_(i) ^(max)))=(2.09 dpt,−0.50 dpt,−0.00 dpt)

This can be used to produce a lens that optimally corrects the mostlikely eye model (e.g. a single vision lens or as a power in thedistance reference point of a progressive lens).

The power of the lens P(

_(i) ^(max)) that optimally corrects the most likely eye model candiffer from the vision disorder of the most likely eye, P^(eye)((

_(i) ^(max))), since the former relates to the corrective lens and thelatter to the power of the eye during refraction, since the distancesbetween the cornea and the corrective lens or refractive lens generallydiffer.

Second Example of the Bayes B Method

The eye model considered in this example as well as the measurement data(here the refraction) are chosen as in the previous second Bayes Aexample. Now, however, the aim is to calculate the most likely power ofan ophthalmic lens to be manufactured, and not the power of theophthalmic lens for the most likely eye model. In this example, theparameters of the ophthalmic lens, L_(i), correspond to their power insimplified form:

L _(i) =P ^(L)=(M ^(L) ,J ₀ ^(L) ,J ₄₅ ^(L)).

For this purpose, power vectors were calculated from the posteriorsample of the previous second Bayes A example, which represent a samplefrom the posterior distribution of the power of the optimal ophthalmiclens (cf. FIGS. 8 a to 8 c ). In comparison to the correspondingdistribution of the density of the powers of the ophthalmic lensresulting from the prior distribution of the eyes (cf. FIGS. 9 a to 9 c), the information gain through the data is clearly recognizable on thebasis of the reduced scattering range of the posterior compared to theprior.

The maximum of the posterior distribution was approximated by themaximum of a kernel density estimation of the sample (multivariatenormal distribution as kernel with a standard deviation corresponding to0.5 times the standard deviation of the posterior distribution of thepower vector). This resulted in the following most likely power of anophthalmic lens to be manufactured (e.g. the power of a spectacle lenssuch as a single vision lens, or the power in the distance referencepoint of a progressive lens), which makes optimal use of the informationavailable about the eye model:

P ^(L, max)=(M ^(L, max) ,J ₀ ^(L, max) ,J ₄₅ ^(L, max))=(1.92 dpt,−0.38dpt,−0.08 dpt).

This most likely power differs from the power P(

_(i) ^(max)) calculated in the Bayes A example for the most likely eyemodel due to the non-linear transformation (here mainly the propagationof the wavefronts between the refracting surfaces) of the parameters ofthe eye model (here the surface powers and distances of the refractivesurfaces and, to a lesser extent, the refractive indices of the media)into the parameters of the ophthalmic lens (here the power of theophthalmic lens as a power vector). As a correction, P^(L, max) in turnrepresents a further improvement compared to the correction P(

_(i) ^(max)) from the second example for the Bayes B method, sinceP^(L, max) corresponds to the most likely corrective power for the giveninformation, and P_(mes) (

_(i) ^(max)) only to the power correcting the most likely eye model butgenerally not the most likely corrective power.

REFERENCE NUMERAL LIST

-   10 main ray-   12 eye-   14 first surface of the spectacle lens (front surface)-   16 second surface of the spectacle lens (back surface)-   18 corneal front surface-   20 eye lens-   30 cutting surface

1.-15. (canceled)
 16. A computer-implemented method for identifyingrelevant individual parameters of at least one eye of a spectacle wearerfor the calculation or optimization of an ophthalmic lens for the atleast one eye of the spectacle wearer, comprising: providing individualdata on properties of the at least one eye of the spectacle wearer;constructing an individual eye model by defining a set of parameters ofthe individual eye model; and determining a probability distribution ofvalues of the parameters of the individual eye model on the basis of theindividual data.
 17. The computer-implemented method according to claim16, wherein: constructing an individual eye model comprises providing aninitial probability distribution of the parameters of the eye model; anddetermining a probability distribution of values of the parameters ofthe individual eye model further takes place on the basis of the initialprobability distribution of parameters of the eye model.
 18. Thecomputer-implemented method according to claim 16, wherein constructingan individual eye model and/or determining a probability distribution ofvalues of the parameters of the individual eye model is carried outusing Bayesian statistics.
 19. The computer-implemented method accordingto claim 16, wherein determining a probability distribution of values ofthe parameters of the individual eye model comprises calculating aconsistency measure, wherein the product of the probability orprobability density of the individual data with given parameters of theindividual eye model with the probability or probability density of theparameters of the individual eye model with given background knowledgeis used as the consistency measure.
 20. The computer-implemented methodaccording to claim 16, further comprising: calculating a probabilitydistribution of parameters of the ophthalmic lens to be calculated oroptimized using at least one parameter of the individual eye model; anddetermining most likely values of the parameters of the ophthalmic lensto be calculated or optimized.
 21. The computer-implemented methodaccording to claim 16, wherein: providing individual data comprisesproviding individual refraction data on the at least one eye of thespectacle wearer; and constricting an individual eye model comprisesdefining an individual eye model in which at least: a shape and/or powerof a corneal front surface of a model eye; and/or a cornea-lensdistance; and/or parameters of the lens of the model eye; and/or alens-retina distance; and/or a size of the entrance pupil; and/or a sizeand/or position of a physical aperture diaphragm is determinable on thebasis of individual measurement values for the eye of the spectaclewearer and/or standard values and/or on the basis of the providedindividual refraction data; and wherein the method further comprises:carrying out a consistency check of the defined eye model with theprovided individual refraction data, and solving any inconsistencieswith the aid of analytical and/or numerical and/or probabilisticmethods.
 22. The computer-implemented method according to claim 21,wherein any inconsistencies are solved by: adapting one or moreparameters of the eye model, wherein several parameters of the eye modelare adapted and the adaptation is divided among the several parametersof the eye model; and/or adding at least one new parameter to the eyemodel and defining it such that the eye model becomes consistent; and/oradapting a target power of the ophthalmic lens.
 23. Acomputer-implemented method for calculating or optimizing an ophthalmiclens for at least one eye of a spectacle wearer, comprising: a methodfor identifying relevant individual parameters of the at least one eyeof the spectacle wearer according to claim 16; specifying a firstsurface and a second surface for the ophthalmic lens to be calculated oroptimized; identifying the course of a main ray through at least onevisual point of at least one surface of the ophthalmic lens to becalculated or optimized into the model eye; evaluating an aberration ofa wavefront resulting from a spherical wavefront incident on the firstsurface of the ophthalmic lens along the main ray on an evaluationsurface compared to a wavefront converging in one point on the retina ofthe eye model; and iteratively varying the at least one surface of theophthalmic lens to be calculated or optimized until the evaluatedaberration corresponds to a predetermined target aberration.
 24. Amethod for producing an ophthalmic lens, comprising: calculating oroptimizing an ophthalmic lens according to the inventive method forcalculating or optimizing an ophthalmic lens according to claim 23; andmanufacturing the thus-calculated or optimized ophthalmic lens.
 25. Adevice for identifying relevant individual parameters of at least oneeye of a spectacle wearer for the calculation or optimization of anophthalmic lens for the at least one eye of the spectacle wearer,comprising: at least one data interface configured to provide individualdata on properties of the at least one eye of the spectacle wearer; anda modeling module configured to model and/or construct an individual eyemodel by defining a set of parameters of the individual eye model,wherein the modeling module is configured to determine a probabilitydistribution of values of the parameters of the individual eye model onthe basis of the individual data.
 26. The device according to claim 25,wherein: providing individual data comprises providing individualrefraction data on the at least one eye of the spectacle wearer; andconstructing an individual eye model comprises defining an individualeye model in which at least: a shape and/or power of a corneal frontsurface of a model eye; and/or a cornea-lens distance; and/or parametersof the lens of the model eye; and/or a lens-retina distance; and/or asize of the entrance pupil; and/or a size and/or position of a physicalaperture diaphragm are defined on the basis of individual measurementvalues for the eye of the spectacle wearer and/or standard values and/oron the basis of the provided individual refraction data, wherein themodeling module is configured to carry out a consistency check of thedefined eye model with the provided individual refraction data and tosolve any inconsistencies with the aid of analytical and/orprobabilistic methods.
 27. A device for calculating or optimizing anophthalmic lens for at least one eye of a spectacle wearer, comprising:a device configured to identify relevant individual parameters of the atleast one eye of the spectacle wearer according to claim 25; a surfacemodel database configured to specify a first surface and a secondsurface for the ophthalmic lens to be calculated or optimized; a mainray identification module configured to identify the course of a mainray through at least one visual point of at least one surface of theophthalmic lens to be calculated or optimized into the model eye; anevaluation module configured to evaluate an aberration of a wavefrontresulting from a spherical wavefront incident on the first surface ofthe ophthalmic lens along the main ray on an evaluation surface comparedto a wavefront converging in one point on the retina of the eye model;and an optimization module configured to iteratively vary the at leastone surface of the ophthalmic lens to be calculated or optimized untilthe evaluated aberration corresponds to a predetermined targetaberration.
 28. A device for producing an ophthalmic lens, comprising: acalculator or optimizer configured to calculate or optimize theophthalmic lens according to a method for calculating or optimizing anophthalmic lens according to claim 23; and a machine configured tomachine the ophthalmic lens in accordance with the result of thecalculation or optimization.
 29. A non-transitory computer programproduct including program code configured to, when loaded and executedon a computer, perform a method for identifying relevant individualparameters of at least one eye of a spectacle wearer according to claim16.
 30. A non-transitory computer program product including program codeconfigured to, when loaded and executed on a computer, perform a methodfor calculating or optimizing an ophthalmic lens according to claim 23.31. A spectacle lens produced by a method according to claim 24.